Definition. 1. HYDRODYNAMICS, from ὕδωρ, "water," and δύναμις, "power," is properly that science which treats of the power of water, whether it acts by pressure or by impulse. In its more enlarged acceptation, however, it treats of the pressure, equilibrium, cohesion, and motion of fluids, and of the machines by which water is raised, or in which that fluid is employed as the first mover. Hydrodynamics is divided into two branches, Hydrostatics and Hydraulics. Hydrostatics comprehends the pressure, equilibrium, and cohesion of fluids, and Hydraulics their motion, together with the machines in which they are chiefly concerned.

HISTORY.

2. The science of hydrodynamics was cultivated with less success among the ancients than any other branch of mechanical philosophy. When the human mind had made considerable progress in the other departments of physical science, the doctrine of fluids had not begun to occupy the attention of philosophers; and, if we except a few propositions on the pressure and equilibrium of water, hydrodynamics must be regarded as a modern science, which owes its existence and improvement to those great men who adorned the seventeenth and eighteenth centuries.

Discoveries of Archimedes. A.C. 250.

3. Those general principles of hydrostatics which are to this day employed as the foundation of that part of the science, were first given by Archimedes in his work Νοητοκείμενον, or De Insidentibus Humido, about 250 years before the birth of Christ, and were afterwards applied to experiments by Marius Getauldus in his Archimedes Proemium. Archimedes maintained that each particle of a fluid mass, when in equilibrium, is equally pressed in every direction; and he inquired into the conditions, according to which a solid body floating in a fluid should assume and preserve a position of equilibrium. We are also indebted to the philosopher of Syracuse for that ingenious hydrostatic process by which the purity of the precious metals can be ascertained, and for the screw engine which goes by his name, the theory of which has lately exercised the ingenuity of some of our greatest mathematicians.

4. In the Greek school at Alexandria which flourished under the auspices of the Ptolemies, the first attempts were made at the construction of hydraulic machinery. About 120 years after the birth of Christ, the fountain of inventions compression, the syphon, and the forcing pump, were invented by Ctesibius and Hero; and though these machines and Hero operated by the elasticity and weight of the air, yet their inventors had no distinct notions of these preliminary branches of pneumatical science. The syphon is a simple instrument which is employed to empty vessels full of water or spirituous liquors, and is of great utility in the arts. The forcing pump, on the contrary, is a complicated and abstruse invention, which could scarcely have been expected in the infancy of hydraulics. It was probably suggested to Ctesibius by the Egyptian school or Noria, Egyptian which was common at that time, and which was a kind of wheel-chain pump, consisting of a number of earthern pots carried round by a wheel. In some of these machines the pots have a valve in their bottom which enables them to descend without much resistance, and diminishes greatly the load upon the wheel; and if we suppose that this valve was introduced so early as the time of Ctesibius, it is not difficult to perceive how such a machine might have led this philosopher to the invention of the forcing pump.

5. Notwithstanding these inventions of the Alexandrian school, its attention does not seem to have been directed to the motion of fluids. The first attempt to investigate this subject was made by Sextus Julius Frontinus, inspector of the public fountains at Rome in the reigns of Nerva and Trajan; and we may justly suppose that his work entitled De Aqueductibus urbis Romae Commentarius contains all the hydraulic knowledge of the ancients. After describing the nine great Roman aqueducts, to which he himself added five more, and mentioning the dates of their erection, he considers the methods which were at that time employed for ascertaining the quantity of water discharged from adjuvatures, and the mode of distributing the waters of an aqueduct or a fountain. He justly remarks that the expense of water from an orifice, depended not only on the magnitude of the orifice itself, but also on the height of the water in the reservoir; and that a pipe employed to carry off a portion of water from an aqueduct, should, as circumstances required, have a position more or less inclined to the original direction of the current. But as he was unacquainted with the true law of the velocities of running water as depending upon the depth of the orifice, we can scarcely be surprised at the want of precision which appears in his results.

It has generally been supposed that the Romans were ignorant of the art of conducting and raising water by means of pipes; but it can scarcely be doubted, from the statement of Pliny and other authors, that they not only were acquainted with the hydrostatical principle, but that they actually used leaden pipes for the purpose. Pliny asserts that water will always rise to the height of its source, and he also adds that, in order to raise water up to an eminence, leaden pipes must be employed.

6. The labours of the ancients in the science of hydrodynamics terminated with the life of Frontinus. The sciences had already begun to decline, and that night of ignorance and barbarism was advancing apace, which for more than a thousand years brooded over the nations of Europe. During this lengthened period of mental degeneracy, when less abstruse studies ceased to attract the notice, and rouse the energies of men, the human mind could not be supposed capable of that vigorous exertion, and patient industry, which are so indispensable in physical researches. Poetry and the fine arts, accordingly, had made considerable progress under the patronage of the family of Medici, before Galileo began to extend the boundaries of science. This great man, who deserves to be called the father and restorer of physics, does not appear to have directed his attention to the doctrine of fluids; but his discovery of the uniform acceleration of gravity, laid the foundation of its future progress, and contributed in no small degree to aid the exertions of genius in several branches of science.

7. Castelli and Torricelli, two of the disciples of Galileo, applied the discoveries of their master to the science of hydrodynamics. In 1628 Castelli published a small work, "Della Misura dell'acque correnti," in which he gave a very satisfactory explanation of several phenomena in the motion of fluids, in rivers and canals. But he committed a great paralogism in supposing the velocity of the water proportional to the depth of the orifice below the surface of the vessel. Torricelli observing that in a jet d'eau where the water rushed through a small adjuvature, it rose to nearly the same height with the reservoir from which it was supplied, imagined that it ought to move with the same velocity as if it had fallen through that height by the force of gravity. And hence he deduced this beautiful and important proposition, that the velocities of fluids are as the square roots of the pressures, abstracting from the resistance of the air and the friction of the orifice. This theorem was published in 1643, at the end of his treatise De Motu Gravium naturaliter accelerato. It was afterwards confirmed by the experiments of Raphael Magliotti, on the quantities of water discharged from different adjuvatures under different pressures; and though it is true only in small orifices, it gave a new turn to the science of hydraulics.

8. After the death of the celebrated Pascal, who discovered the pressure of the atmosphere, a treatise on the equilibrium of fluids (Sur l'Equilibre des Liquides), was found among his manuscripts, and was given to the public in 1663. In the hands of Pascal, hydrostatics assumed the dignity of a science. The laws of the equilibrium of fluids were demonstrated in the most perspicuous and simple manner, and amply confirmed by experiments. The discovery of Torricelli, it may be supposed, would have incited Pascal to the study of hydraulics. But as he has not treated this subject in the work which has been mentioned, it was probably composed before that discovery had been made public.

9. The theorem of Torricelli was employed by many succeeding writers, but particularly by the celebrated Mariotte, whose labours in this department of physics deserve to be recorded. His Traité du Mouvement des Eaux, which was published after his death in the year 1686, is founded on a great variety of well-conducted experiments on the motion of fluids, performed at Versailles and Chantilly. In the discussion of some points he has committed considerable mistakes. Others he has treated very superficially, and in none of his experiments does he seem to have attended to the diminution of efflux arising from the contraction of the fluid vein, when the orifice is merely a perforation in a thin plate; but he appears to have been the first who attempted to ascribe the discrepancy between theory and experiment to the retardation of the water's velocity arising from friction. His contemporary Guglielmini, who was inspector of the rivers and canals in the Milanese, had ascribed this diminution of velocity in rivers, to transverse motions arising from inequalities in their bottom. But as Mariotte observed similar obstructions even in glass pipes, where no transverse currents could exist, the cause assigned by Guglielmini seemed destitute of foundation. The French philosopher, therefore, regarded these obstructions as the effects of friction. He supposes that the filaments of water which graze along the sides of the pipe lose a portion of their velocity; that the contiguous filaments having on this account a greater velocity, rub upon the former, and suffer a diminution of their celerity; and that the other filaments are affected with similar retardations proportional to their distance from the axis of the pipe. In this way the medium velocity of the current may be diminished, and consequently the quantity of water discharged in a given time, must, from the effects of friction, be considerably less than that which is computed from theory.

10. That part of the science of hydrodynamics which relates to the motion of rivers seems to have originated in Italy. This fertile country receives from the Apennine rivers first a great number of torrents, which traverse several principalities before they mingle their waters with those of the Po, into which the greater part of them fall. To defend themselves from the inundations with which they were threatened, it became necessary for the inhabitants to change the course of their rivers; and while they thus drove them from their own territories, they let them loose on those of their neighbours. Hence arose the continual quarrels which once raged between the Bolognese and the inhabitants of Modena and Ferrara. The attention of the Italian engineers was necessarily directed to this branch of science; and from this cause a greater number of works... History. were written on the subject in Italy than in all the rest of Europe.

11. Guglielmini was the first who attended to the motion of water in rivers and open canals. Embracing the theorem of Torricelli, which had been confirmed by repeated experiments, Guglielmini concluded that each particle in the perpendicular section of a current has a tendency to move with the same velocity as if it issued from an orifice at the same depth from the surface. The consequences deducible from this theory of running waters are in every respect repugnant to experience, and it is really surprising that it should have been so hastily adopted by succeeding writers. Guglielmini himself was sufficiently sensible that his parabolic theory was contrary to fact, and endeavoured to reconcile them by supposing the motion of rivers to be obstructed by transverse currents arising from irregularities in their bed. The solution of this difficulty, as given by Mariotte, was more satisfactory, and was afterwards adopted by Guglielmini, who maintained also that the viscosity of water had a considerable share in retarding its motion.

12. The effects of friction and viscosity in diminishing the velocity of running water were noticed in the Principia of Sir Isaac Newton, who has thrown much light upon several branches of hydrodynamics. At a time when the Cartesian system of vortices universally prevailed, this great man found it necessary to investigate that absurd hypothesis, and in the course of his investigations he has shown that the velocity of any stratum of the vortex is an arithmetical mean between the velocities of the strata which enclosed it; and from this it evidently follows, that the velocity of a filament of water moving in a pipe is an arithmetical mean between the velocities of the filaments which surround it. Taking advantage of these results, it was afterwards shewn by M. Pitot, that the retardations arising from friction are inversely as the diameters of the pipes in which the fluid moves. The attention of Newton was also directed to the discharge of water from orifices in the bottom of vessels. He supposed a cylindrical vessel full of water to be perforated in its bottom with a small hole by which the water escaped, and the vessel to be supplied with water in such a manner that it always remained full at the same height. He then supposed this cylindrical column of water to be divided into two parts; the first, which he calls the cataract, being a hyperboid generated by the revolution of a hyperbola of the fifth degree around the axis of the cylinder which should pass through the orifice; and the second the remainder of the water in the cylindrical vessel. He considered the horizontal strata of this hyperboloid as always in motion, while the remainder of the water was in a state of rest; and imagined that there was a kind of cataract in the middle of the fluid. When the results of this theory were compared with the quantity of water actually discharged, Newton concluded that the velocity with which the water issued from the orifice was equal to that which a falling body would receive by descending through half the height of water in the reservoir. This conclusion, however, is absolutely irreconcilable with the known fact, that jets of water rise nearly to the same height as their reservoirs, and Newton seems to have been aware of this objection. In the second edition of his Principia, accordingly, which appeared in 1714, Sir Isaac has reconsidered his theory. He had discovered a contraction in the vein of fluid (vena contracta), which issued from the orifice, and found that, at the distance of about a diameter of the aperture, the section of the vein was contracted in the subduplicate ratio of two to one. He regarded, therefore, the section of the contracted vein as the true orifice from which the discharge of water ought to be deduced, and the velocity of the issuing fluid was equal to that acquired by falling through D. By adapting to a circular orifice through which the water escaped, a cylindrical tube of the same diameter, the Marquis found that the quantity discharged in a determinate time was considerably greater than when it issued from the circular orifice itself; and this happened whether the water descended perpendicularly or issued in a horizontal direction.

15. Such was the state of hydrodynamics in 1738, when Daniel Bernoulli published his Hydrodynamicae, seu de vi fluidorum Commentarii. His theory of its theory of the motion of fluids was founded on two suppositions, which appeared to him conformable to experience. He supposed first, that the surface of a fluid, contained in a vessel which was emptying itself by an orifice, remains always horizontal; and if the fluid mass is conceived to be divided into an infinite number of horizontal strata of the same bulk, that these strata remain contiguous to each other, and that all their points descend vertically, with velocities inversely proportional to their breadth, or to the horizontal sections of the reservoir. In order to determine the motion of each stratum, he employed the principle of the conservatio virt-

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1 See his principal work, entitled La Misura dell' acqua corrente. History. um vivarium, and obtained very elegant solutions. In the opinion of the Abbé Bossut, his work was one of the finest productions of mathematical genius.

16. The uncertainty of the principle employed by Daniel Bernoulli, which has never been demonstrated in a general manner, deprived his results of that confidence which they would otherwise have deserved; and rendered it desirable to have a theory more certain, and depending solely on the fundamental laws of mechanics. Maclaurin and John Bernoulli, who were of this opinion, resolved the problem by more direct methods, the one in his Fluxions, published in 1742; and the other in his Hydrodynamica nunc primum detecta, et directe demonstrata ex principiis purè mechanicis, which forms the fourth volume of his works. The method employed by Maclaurin has been thought not sufficiently rigorous; and that of John Bernoulli is, in the opinion of La Grange, defective in perspicuity and precision.

17. The theory of Daniel Bernoulli was opposed also by the celebrated D'Alembert. When generalising James Bernoulli's Theory of Pendulums, he discovered a principle of dynamics so simple and general, that it reduced the laws of the motions of bodies to that of their equilibrium. He applied this principle to the motion of fluids, and gave a specimen of its application at the end of his Dynamics in 1743. It was more fully developed in his Traité des Fluides, which was published in 1744, where he has resolved, in the most simple and elegant manner, all the problems which relate to the equilibrium and motion of fluids. He makes use of the very same suppositions as Daniel Bernoulli, though his calculus is established in a very different manner. He considers, at every instant, the actual motion of a stratum, as composed of a motion which it had in the preceding instant, and of a motion which it has lost. The laws of equilibrium between the motions lost, furnish him with equations which represent the motion of the fluid. Although the science of hydrodynamics had then made considerable progress, yet it was chiefly founded on hypothesis. It remained a desideratum to express by equations the motion of a particle of the fluid in any assigned direction. These equations were found by D'Alembert, from two principles, that a rectangular canal, taken in a mass of fluid in equilibrium, is itself in equilibrium; and that a portion of the fluid, in passing from one place to another, preserves the same volume when the fluid is incompressible, or dilates itself according to a given law when the fluid is elastic. His very ingenious method was published in 1752, in his Essai sur la résistance des fluides. It was brought to perfection in his Opuscules Mathématiques, and has been adopted by the celebrated Euler.

Before the time of D'Alembert, it was the great object of philosophers to submit the motion of fluids to general formulae, independent of all hypothesis. Their attempts, however, were altogether fruitless; for the method of fluxions, which produced such important changes in the physical sciences, was but a feeble auxiliary in the science of hydraulics. For the resolution of the questions concerning the motion of fluids, we are indebted to the method of partial differences, a new calculus, with which Euler enriched the sciences. This great discovery was first applied to the motion of water by the celebrated D'Alembert, and enabled both him and Euler to represent the theory of fluids in formulæ restricted by no particular hypothesis.

18. An immense number of experiments on the motion of water in pipes and canals was made by Professor Michelotti of Turin, at the expense of the sovereign. In these experiments the water issued from holes of different sizes, under pressures of from 5 to 22 feet, from a tower constructed of the finest masonry. Basins (one of which was 289 feet square) built of masonry, and lined with stucco, received the effluent water, which was conveyed in canals of brickwork, lined with stucco, of various forms and declivities. The whole of Michelotti's experiments were conducted with the utmost accuracy; and his results, which are in every respect entitled to our confidence, were published in 1774 in his Sperienze Idrauliche.

19. The experiments of the Abbé Bossut, whose labours in this department of science have been very assiduous and successful, have, in as far as they coincide, afforded the same results as those of Michelotti. Though performed on a smaller scale, they are equally entitled to our confidence, and have the merit of being made in cases which are most likely to occur in practice. In order to determine what were the motions of the fluid particles in the interior of a vessel emptying itself by an orifice, M. Bossut employed a glass cylinder, to the bottom of which different adjutages were fitted; and he found that all the particles descend at first vertically, but that at a certain distance from the orifice they turn from their first direction towards the aperture. In consequence of these oblique motions, the fluid vein forms a kind of truncated conoid, whose greatest base is the orifice itself, having its altitude equal to the radius of the orifice, and its bases in the ratio of 3 to 2.—It appears also, from the experiments of Bossut, that when water issues through an orifice made in a thin plate, the expense of water, as deduced from theory, is to the real expense as 16 to 10, or as 8 to 5; and, when the fluid issues through an additional tube, two or three inches long, and follows the sides of the tube, as 16 to 13.—In analyzing the effects of friction, he found, 1. That small orifices gave less water in proportion than great ones, on account of friction; and, 2. That when the height of the reservoir was augmented, the contraction of the fluid vein was also increased, and the expense of water diminished; and by means of these two laws he was enabled to determine the quantity of water discharged, with all the precision he could wish. In his experiments on the motion of water in canals and tubes, he found that there was a sensible difference between the motion of water in the former and in the latter. Under the same height of reservoir, the same quantity of water always flows in a canal, whatever be its length and declivity; whereas, in a tube, a difference in length and declivity has a very considerable influence on the quantity of water discharged. According to the theory of the resistance of fluids, the impulse upon a plane surface is as the product of its area multiplied by the square of the fluid's velocity, and the square of the sine of the angle of incidence. The experiments of Bossut, made in conjunction with D'Alembert and Condorcet, prove, that this is sensibly true when the impulse is perpendicular; but that the aberrations from theory increase with the angle of impulsion. They found, that when the angle of impulsion was between 50° and 90°, the ordinary theory may be employed, that the resistances thus found will be a little less than they ought to be, and the more so as the angles recede from 90°. The attention of Bossut was directed to a variety of other interesting points, which we cannot stop to notice, but for which we must refer the reader to the works of that ingenious author.

20. The oscillation of waves, which was first discussed inquiries by Sir Isaac Newton, and afterwards by D'Alembert, in de Flauger's article Ondes in the French Encyclopædia, was now revived by M. Flaugergues, who attempted to overthrow the conclusions of these philosophers. He maintained, that a motion of wave is not the effect of a motion in the particles of water, waves.

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1 The germ of Daniel Bernoulli's theory was first published in his memoir entitled Theoria Novae de Motu Aquarum per Canales quæque Fluente, which he had communicated to the Academy of St. Petersburg as early as 1726. by which they rise and fall alternately, in a serpentine line, when moving from the centre where they commenced; but that it is a kind of intumescence, formed by a depression at the place where the impulse is first made, which propagates itself in a circular manner when removing from the point of impulse. A portion of the water, thus elevated, he imagines, flows from all sides into the hollow formed at the centre of impulse, so that the water being, as it were, heaped up, produces another intumescence, which propagates itself as formerly. From this theory M. Flaugergues concludes, and he has confirmed the conclusion by experiment, that all waves, whether great or small, have the same velocity.

And of M. Lagrange. Lagrange, in his Mécanique Analytique. He found that the velocity of waves in a canal is equal to that which a heavy body would acquire by falling through a height equal to half the depth of the water in the canal. If this depth, therefore, be one foot, the velocity of the waves will be 5'945 feet in a second; and if the depth is greater or less than this, their velocity will vary in the subduplicate ratio of the depth, provided it is not very considerable. If we suppose that, in the formation of waves, the water is agitated but to a very small depth, the theory of Lagrange may be employed, whatever be the depth of the water and the figure of its bottom. This supposition, which is very plausible, when we consider the tenacity and adhesion of the particles of water, has also been confirmed by experience.

The most successful labourer in the science of hydrodynamics was the Chevalier Dubuat, engineer in ordinary to the King of France. Following in the steps of the Abbé Bossut, he prosecuted the inquiries of that philosopher with uncommon ingenuity; and in the year 1786 he published, in two volumes, his Principes d'Hydraulique, which contains a satisfactory theory of the motion of fluids, founded solely upon experiments. The Chevalier Dubuat considered that if water were a perfect fluid, and the channels in which it flowed infinitely smooth, its motion would be continually accelerated, like that of bodies descending in an inclined plane. But as the motion of rivers is not continually accelerated, and soon arrives at a state of uniformity, it is evident that the viscosity of the water, and the friction of the channel in which it descends, must equal the accelerating force. M. Dubuat, therefore, assumes it as a proposition of fundamental importance, that when water flows in any channel or bed, the accelerating force, which obliges it to move, is equal to the sum of all the resistances which it meets with, whether they arise from its own viscosity or from the friction of its bed. This principle was employed by M. Dubuat, in the first edition of his work, which appeared in 1779; but the theory contained in that edition was founded on the experiments of others. He soon saw, however, that a theory so new, and leading to results so different from the ordinary theory, should be founded on new experiments more direct than the former, and he was employed in the performance of these from 1780 to 1783. The experiments of Bossut having been made only on pipes of a moderate declivity, M. Dubuat found it necessary to supply this defect. He used declivities of every kind, from the smallest to the greatest; and made his experiments upon channels, from a line and a half in diameter, to seven or eight square toises.

M. Venturi, Professor of Natural Philosophy in the University of Modena, succeeded in bringing to light some curious facts respecting the motion of water, in his work on the Lateral Communication of Motion in Fluids. He observed, that if a current of water is introduced with a certain velocity into a vessel filled with the same fluid at rest, and if this current, passing through a portion of the fluid, is received in a curvilinear channel, the bottom of which gradually rises till it passes over the rim of the vessel itself, it will carry along with it the fluid contained in the vessel; so that after a short time has elapsed, there remains only the portion of the fluid which was originally below the aperture at which the current entered. This phenomenon has been called by Venturi, the lateral communication of motion in fluids; and, by its assistance, he has endeavoured to explain many important facts in hydraulics. He has not attempted to explain this principle; but has shown that the mutual action of the fluid particles does not afford a satisfactory explanation of it. The work of Venturi contains many other interesting discussions, which are worthy of the attention of every reader.

Although the Chevalier Dubuat had shown much sagacity in classifying the different kinds of resistances which he supposed to be exhibited in the motion of fluids, yet it was reserved for Coulomb to express the sum of them by a rational function of the velocity. By a series of interesting experiments on the successive diminution of the oscillation of disks, arising from the resistance of the water in which they oscillated, he was led to the conclusion, that the pressure sustained by the moving disk is represented by two terms, one of which varies with the simple velocity, and the other with its square. When the motions are very slow, the part of the resistance proportional to the square of the velocity is insensible, and hence the resistance is proportional to the simple velocity. M. Coulomb found also, that the resistance is not perceptibly increased by increasing the depth of the oscillating disk in the fluid; and by coating the disk successively with fine and coarse sand, he found that the resistance arises solely from the mutual cohesion of the fluid particles, and from their adhering to the surface of the moving body.

The law of resistance discovered by Coulomb, was first applied to the determination of the velocity of running water by M. Girard, who considers the resistance as represented by a constant quantity, multiplied by the sum of the first and second powers of the velocity. He regards the water which moves over the wetted sides of the channel as at first retarded by its viscosity, and he concludes that the water will, from this cause, suffer a retardation proportional to the simple velocity. A second retardation, analogous to that of friction in solids, he ascribes to the roughness of the channel, and he represents it by the second power of the velocity, as it must be in the compound ratio of the force and the number of impulsions which the asperities receive in a given time. He then expresses the resistance due to cohesion by a constant quantity, to be determined experimentally, multiplied into the product of the velocity of the perimeter of the section of the fluid.

The influence of heat in promoting fluidity was known to the ancients; but M. Dubuat was the first person who investigated the subject experimentally. His results, however, were far from being satisfactory; and it was left to M. Girard to ascertain the exact effect of temperature on the motion of water in capillary tubes. When the length of the capillary tube is great, the velocity is quadrupled by an increase of heat from 0° to 85° centigr.; but when its length is small, a change of temperature exercises little or no influence on the velocity. He found also, that, in ordinary conduit pipes, a variation of temperature exercises scarcely any influence over the velocity.

The theory of running water was greatly advanced by the researches of M. Prony. From a collection of the best experiments by Couplet, Bossut, and Dubuat, he selected 82, of which 51 were made on the velocity of water. in conduit pipes, and 31 on its velocity in open canals; and by discussing these on physical and mechanical principles, he succeeded in drawing up general formulas which afford a simple expression of the velocity of running water.

28. M. Eykelwein of Berlin published, in 1801, a valuable compendium of Hydraulics, entitled Handbuch der Mechanik und der Hydraulik, which contains an account of many new and valuable experiments made by himself. His work is divided into 24 chapters, the most important of which are the 7th, which treats of the motion of water in rivers, and the 9th, which treats of the motion of water in pipes. He has shown that the mean velocity of water in a second in a river or canal flowing in an equable channel, is 49ths of a mean proportional between the fall in two English miles, and the hydraulic mean depth; and that the superficial velocity of a river is nearly a mean proportional between the hydraulic mean depth and the fall in two English miles.

The following are some of the other important results which are given in his work:—The contraction of the fluid area is 064, the coefficient for additional pipes 065, the coefficient for a conical tube similar to the curve of contraction 098. For the whole velocity due to the height, the coefficient by its square must be multiplied by 80458. For an orifice, the coefficient must be multiplied by 78; for wide openings in bridges, sluices, &c., by 69; for short pipes, 69; and for openings in sluices without side walls, 51. Our author investigates the subject of the discharge of water by compound pipes, the motions of jets, and their impulses against plane and oblique surfaces, and he shows theoretically, that a water wheel will have its effect a maximum when its circumference moves with half the velocity of the stream.

29. A series of interesting hydraulic experiments was made at Rome in 1809, by MM. Mallet and Vici. They found that a pipe, whose gauge was five ounces French measure (or 003059 French kilolitres), furnished one-seventh more water than five pipes of one ounce, an effect arising from the velocity being diminished by friction in the ratio of the perimeters of the orifices as compared with their areas.

Notwithstanding the investigations of Newton, D'Alembert, and Lagrange, the problem of waves was still unsolved; and the Institute of France was induced to propose, as the subject of its annual prize for 1816, "The Theory of Waves on the Surface of a heavy Fluid of indefinite Depth." M. Poisson had previously studied this difficult subject, and he lodged his first memoir in the bureau of the Institute on the 2d October 1815, at the expiration of the period allowed for competition. M. Poisson supposes the waves to be produced in the following manner:—A body of the form of an elliptic paraboloid is immersed a little in the fluid, with its axis vertical and its vertex downwards. After being left in this position till the equilibrium of the fluid is restored, the body is suddenly withdrawn, and waves are formed round the place which it occupied. This first memoir contains the general formula for waves propagated with a uniformly accelerated motion; but in a second memoir, read in December, he gives the theory of waves propagated with a constant velocity. This last class of waves are much more sensible than the first, and are those which are seen to spread in circles round any disturbance made at the surface of water. In determining the superficial as well as the internal propagation of these waves, he considers only the case when the disturbance of the water is so small, that the second and the higher powers of the velocity of the oscillating particles may be neglected; and he assumes, that a fluid particle which is at any instant at the surface, continues there during the whole of the motion, a supposition which the condition of the continuity of the fluid renders necessary. He supposes the depth of the water constant throughout its whole extent, the bottom being considered as a fixed horizontal plane at a given distance beneath its natural surface. He then treats, first, the case in which the motion takes place in a canal of uniform width, over which obstruction is made of the horizontal dimension of the fluid; and, secondly, the case in which the fluid is considered in its true dimensions.

30. The prize offered by the Institute was gained by M. Augustin Louis Cauchy, then a young mathematician of M. Cauchy, the highest promise. In his memoir, which was published in the third volume of the Mémoires des Sciences, he treats only of the first kind of waves above mentioned; and his investigation claims to be more complete than that in the first memoir of Poisson, in so far as it leaves entirely arbitrary the form of the function relative to the initial form of the fluid surface, and, therefore, allows the analysis to be applied when bodies of different forms are used to produce the initial disturbance. From his analysis, M. Cauchy concludes, "that the heights and velocities of the different waves produced by the immersion of a cylindrical or prismatic body, depend not only on the width and height of the part immersed, but also on the form of the surface which bounds this part." He is also of opinion, that the number of the waves produced may depend on the form of the immersed body, and the depth of immersion.

31. The following abstract of the principal results obtained by theory, respecting the nature of waves, has been given by Mr Challis:

"With respect first to the canal of uniform width, the law of the velocity of propagation found by Lagrange, is confirmed by Poisson's theory when the depth is small, but not otherwise.

"When the canal is of unlimited depth, the following are the chief results:

"1. An impulse given to any point of the surface, affects instantaneously the whole extent of the fluid mass.

"The theory determines the magnitude and direction of the initial velocity of each particle resulting from a given impulse.

"2. The summit of each wave moves with a uniformly accelerated motion.

"This must be understood to refer to a series of very small waves, called by M. Poisson dents, which perform their movements, as it were, on the surface of the larger waves, which he calls 'les ondes dentelles.' Each wave of the series is found to have its proper velocity, independent of the primitive impulse.

"3. At considerable distances from the place of disturbance, there are waves of much more sensible magnitude than the preceding.

"Their summits are propagated with a uniform velocity, which varies as the square root of the breadth, à fleur d'eau of the fluid originally disturbed. Yet the different waves which are formed in succession, are propagated with different velocities; the foremost travels swiftest. The amplitudes of oscillations of equal duration, are reciprocally proportional to the square root of the distances from the point of disturbance.

"4. The vertical excursions of the particles situated directly below the primitive impulse, vary according to the inverse ratio of the depth below the surface. This law of decrease is not so rapid but that the motion will be very sensible at very considerable depths; it will not be the true law, as the theory proves, when the original disturbance extends over the whole surface of the water, for the decrease of motion in this case will be much more rapid.

"The results of the theory, when the three dimensions of the fluid are considered, are analogous to the preceding,

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1 See Mallet, Notices Historiques, Paris, 1830. History. 1, 2, 3, 4, and may be stated in the same terms, excepting that the amplitudes of the oscillations are inversely as the distances from the origin of disturbance, and the vertical excursions of the particles situated directly below the disturbance vary inversely as the square of the depth."

32. M. Hachette, in the year 1816, presented to the National Institute a memoir containing the results of experiments which he had made on the spouting of fluids, and the discharge of vessels. The objects he had in view were to measure the contracted part of a fluid vein; to examine the phenomena attendant on additional tubes; and to investigate and describe the figure of the fluid vein, and the results which take place when different forms of orifices are employed. M. Hachette showed in the second part of his memoir, that greater or lesser volumes of water will be discharged in the same time through tubes of different figures, the apertures in all having the same dimensions. He also gave several curious results respecting other fluids issuing out of orifices into air or a vacuum.

33. Several very interesting experiments on the propagation of waves, have been made by M. Weber and by M. Bidone. Although Weber's experiments were not made in exact conformity with the condition which the theory required, yet they, generally speaking, harmonize with it; and they particularly establish the existence of the small accelerated waves near the place of disturbance, and of a perceptible motion of the particles of the fluid at considerable depths below the surface. When an elliptic paraboloid is used to produce the waves, with its axis vertical, and its vertex downwards, and when, of course, the section of the solid in the plane of the surface of the water is an ellipse, the velocity of propagation is, according to the theory, greater in the direction of the major axis than in that of the minor, in the ratio of the square root of the one to the square root of the other; but this result is not confirmed by Weber's experiments.

In 1826 M. Bidone, besides his experiments on waves, made a series on the velocity of running water at the hydraulic establishment of the University of Turin, and he published an account of them in 1829. After giving a description of his apparatus and method of experimenting, he gives the figures obtained from fluid veins, sections of which were taken at different distances from the orifice, and the results, which are extremely curious, are illustrated by diagrams.

34. Towards the end of last, and the beginning of the present century, several hydraulic engines were invented, which have proved of great service to the arts. The Bramah press was patented by its inventor in 1796. Like all inventions of the first class, it has received little if any improvement since it left Bramah's hand. It derives its great power from the principle of fluids transmitting pressure in every direction. In the year 1775 Mr Whitehurst devised a machine for elevating water, the principle of its action being the momentum of the water itself. This machine was in 1798 much improved by Mr Boulton; it is now, however, replaced by the more scientific invention known as Montgolfier's hydraulic ram, which is recognised by some as an independent invention, but by others as an improvement on Whitehurst's. About the year 1817, Hall's machines began to be revived and greatly improved by M. Reichenbach; in his hands they were transformed into the elegant yet powerful water-column machines, so extensively employed in the German and French mines.

35. M. Poisson's first memoir on the Theory of Capillary Attraction, was read before the National Institute, November 1828, in which his object was to form the equations of equilibrium of fluids on physical principles, that is, by supposing that a fluid mass is made up of distinct molecules, separated from each other by spaces excessively small, and void of ponderable matter. His second memoir, entitled Nouvelle Théorie de l'Action Capillaire, was published in May 1834. The principal object which Poisson had in view in this treatise was to bring the theory of capillary attraction to the greatest degree of perfection that the power of analysis and the knowledge of facts would admit. Laplace's theory of capillarity, however, is still used to explain the phenomena of the subject.

36. A series of useful experiments was made in 1827 and Expériences of Weber and Bidone, 1825. under the sanction of the French government, by means of MM. Poncelet and Lesbros, at the Military School of Metz. The apparatus which they used consisted of four basins, from the first had an area of 25,000 sq. met.; the second one of 1827-28, 1500 sq. met., and a depth of 3-70 met., and was so contrived, by means of sluices, as to have a complete command of the level of the water during the experiment; the third basin, which communicated directly with the second basin, was 365 mct. long and 3 wide, to receive the water discharged by the orifices; and the fourth basin was a gauge capable of containing 24,000 litres. In carrying on the experiments, the opening of the orifices, the height or charge of the fluid in the reservoir, as well as the level of the water in the gauge basin relative to each discharge of fluid, were measured to the tenth of a millimetre, so that the approximation was at least \( \frac{1}{10} \)th of the whole result.

These experiments were made on rectangular vertical orifices 20 centimetres wide, pierced in a thin plane wall, and completely isolated both from the sides and bottom of the vessel. The special object which Poncelet and Lesbros had in view was to determine the laws of escape of water from such orifices, limited towards the upper edge by a moveable flood-gate:—1. In the hypothesis of thin plates, where, on the water escaping freely into the air, the orifice is completely isolated from the side walls and bottom of the reservoir; 2. In the hypothesis where the orifice is more or less near the bottom and sides, and disposed perpendicularly or obliquely with respect to the plate containing it; 3. In the hypothesis of thick walls, where the water would be immediately received into a course or canal of short length, open at the upper edge; and, 4. To study the physical or mathematical laws of each phenomenon, and to inquire into the causes why the results of experiment differ from those of theory. The experimenters never lost sight of the application of these laws to practice.

In the year 1827 M. Poncelet published a Mémoire sur les Roues Hydrauliques à Aubes Courbes, containing his experiments on the undershot wheel with curved palettes, which he had invented in the year 1824. The best undershot previous to the introduction of the Poncelet wheel, never developed more than 0-25 of the work of the water, whereas this utilized 0-60 of that work, which is nearly equivalent to the maximum effect of the breast wheel. The principle on which the Poncelet wheel acts, and that which makes it utilize so much of the work of the water, is that the water on entering any curved palette, enters it without any shock, and continues in this state, till at last, when about to quit the palette, it does so without any sensible velocity. This under-shot wheel is much used in France.

Previous to the year 1827, the wheels required in the mills and manufactories of Germany and France were generally those which worked with the axis horizontal, or the tub and spoon wheels with the axis vertical; but in that year a young mechanic named Fourneryon introduced a wheel working with the axis vertical, yet wholly different from the

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1 Wellenlehre auf Experimente gegründet, Leipzig, 1825. 2 Turin Memoire, vol. xxxv. 3 Expériences sur la forme et sur la direction des Veines et des Courans d'Eau, lancés par diverses ouvertures, 1832. 4 Expériences Hydrauliques sur les Lois de l'Ecoulement de l'Eau, etc., 1832. latter kind. Instead of introducing this horizontal wheel within, as was at that time done, he placed it without the cylinder, opposite its lower part, and separated from it by a small space. In order to put the wheel in motion, water was introduced into the cylinder from above, and escaped by a circular opening at its lower part opposite the wheel. As the cylinder is always kept full of water, the palettes are conveniently struck, and all at once. From this simple arrangement, it results that this hydraulic engine, or turbine, as it has been called, is the best in use. Shortly after the invention was made public, M. Fourneyron was awarded the prize of 6000 francs, which was offered by the Society for the Encouragement of National Industry. These turbine wheels are mostly used on the Continent; they utilize between 75 and 85 per cent. of the work expended by the water.

Mr George Rennie undertook a series of experiments on the friction of water against revolving cylinders and disks, on the direct resistances to globes and disks revolving in air and water alternately, on the coefficients of contraction, and on the expenditure of water through orifices, additional tubes, and pipes of different lengths.

Mr J. Scott Russell's experiments on the motion of vessels and on waves, and the consequences he deduces therefrom, are to be seen in his Researches in Hydrodynamics, 1837.

The late Mr Jardine made several experiments on the discharge of water from long pipes. The waste actually given by the supply pipe from Comiston to the reservoir on the Castle Hill, Edinburgh, was compared with the volume as given by the formulae of Eytelwein, Gerard, Dubuat, and Prony, and he found that the two were nearly coincident.

An improvement on the common theory of fluids was lately suggested by Professor Airy, in his lectures at Cambridge. It had been usual to assume the law of equal pressure as a datum of observation. Professor Airy, however, has shown that this property may be deduced from another more simple, and equally given by observation; namely, that the division of a perfect fluid may be effected without the application of sensible force; and hence, it immediately follows, that the state of equilibrium or motion of a mass of fluid, is not altered by a mere separation of its parts by an indefinitely thin partition. Professor Miller has given a definition of fluids founded on this principle, and a proof of the law of equal pressure, at the beginning of his Elements of Hydrostatics, &c., published at Cambridge in 1831. Dr Thomas Young had employed an equivalent principle to that of Mr Airy in determining the manner of the reflection of waves of water; and Mr Challis considers it as necessary in the solution of some hydrostatal and hydrodynamical problems.

There are certain cases in the analytical theory of Hydrodynamics which require a more simple analysis than others; such, for example, as those of steady motion, or of motion which has arrived at a permanent state, so that the velocity is constant in quantity and direction at the same point. Equations applicable to this kind of motion seem to have been first given by Professor Mosely, in his Elementary Treatise on Hydrostatics and Hydrodynamics. The following is the principle from which he has derived them. When the motion is steady, each particle, in passing from one point to another, passes successively through the states of motion of all the particles which at any instant lie in its path. This general principle is applicable to all kinds of fluids, and is true, whether or not the effect of heat is taken into account, provided the condition of steadiness remains. As it enables us to consider the motion of a single particle in place of that of a number, it readily affords the equations of motion.

As it was desirable to know how the same equation might be obtained from the general equations of fluid motion, Mr Challis undertook this inquiry, and published the results of it in the Transactions of the Cambridge Philosophical Society. In this paper, he has given a method of doing this both for incompressible and elastic fluids, and he has shown that a term in the general formulae which occasions the complexity in most hydrodynamical questions, disappears in cases of steady motion. Mr Challis is of opinion that these equations may be employed in very interesting researches, and he mentions, as instances, the motion of the atmosphere as affected by the rotation of the earth, and a given distribution of temperature due to solar heat.

The science of Hydrodynamics has of late years been experimented with by M. Eytelwein of Berlin, by Dr Matthew Young, who has explained the cause of the increased velocity of efflux through additional tubes, by Mr Vince, Dr T. others, Young, Coulomb, and Don George Juan.

The subject of weirs has occupied the attention of Mr Green, Professor Kelland, Mr Earnshaw, and more especially Mr Airy, the astronomer-royal, who has given much valuable and original matter in his excellent treatise on Tides and Waves. Our knowledge of the flow of water through orifices and over weirs has been much extended by the experiments of MM. Daubisson and Castel, undertaken between the years 1830 and 1838. Mr Provis, in 1838, experimented on the flow of water through small pipes. About the year 1840, M. V. Regnault instituted a series of interesting experiments, by order of the minister of public works, and at the request of the central commission of steam engines, for the purpose of determining the principal laws and numerical data which enter into the calculation of steam engines. The nature of his investigations necessarily led him to determine the compressibility of water and mercury, and his results may be considered as very accurate. The experiments are detailed at great length in the Memoires de l'Institut for 1847.

Professor Magnus of Berlin, in 1848, made several experiments on the motion of fluids; they were intended to refute the notion of Venturi, who, in his Lateral Communication of Motion in Fluids, asserts that if any very mobile body be caused to approach a jet of water, the body will be carried forward by the air, to which a motion has been imparted by the jet. He next shows that the action of a jet on a plane, the jet and plane being under the surface of water, increases with the distance from the orifice; and, lastly, he attempted to ascertain the manner in which the water of the jet mixes with that into which it is projected. Professor Challis has considered the Principles of Hydrodynamics, in the new series of the Philosophical Magazine, 1851. Mr T. E. Blackwell made in 1851 a very extensive and practically useful series of experiments on weirs. The last matter that we shall mention in this history of the subject, is the centrifugal pump, improved by Mr Appold. Bramah and Dietz long ago attempted to produce machines working by a continuous movement of rotation. Dietz' centrifugal pump has been a considerable time in use; it is far surpassed by Appold's Centrifugal Pump. Mr Appold received for his invention the council medal at the Great Exhibition of 1851. Mr Appold was the first who successfully applied curved vanes to the centrifugal pump. It appears, however, that so far back as 1839, curved vanes were adopted in pumps erected in the United States, but for some reason or other, they were replaced by plain straight fins, and therefore of less efficiency. Gwinne, Bessemer, and several others, have made improvements on the centrifugal pump.

We refer the reader to M. Ch. Combe's Sur les Roues à Réaction ou à Turbines, 1848; and to a historical review of the Centrifugal Pump in the Practical Mechanic's Magazine for 1851. PART I.—HYDROSTATICS.

41. Hydrostatics is that branch of the science of hydrodynamics which comprehends the pressure and equilibrium of non-elastic fluids, as water, oil, mercury, &c.; the method of determining the specific gravities of substances, the equilibrium of floating bodies, and the phenomena of capillary attraction.

DEFINITIONS AND PRELIMINARY OBSERVATIONS.

42. A fluid is a collection of very minute particles, cohering so little among themselves, that they yield to the smallest force, and are easily moved among one another.

43. Fluids have been divided into perfect and imperfect. In perfect fluids the constituent particles are supposed to be endowed with no cohesive force, and to be moved among one another by a pressure infinitely small. But, in imperfect or viscous fluids, the mutual cohesion of their particles is very sensible, as in oil, varnish, melted glass, &c.; and this tenacity prevents them from yielding to the smallest pressure. Although water, mercury, alcohol, &c. have been classed among perfect fluids, yet it is evident that neither these nor any other liquid is possessed of perfect fluidity. When a glass vessel is filled with water above the brim, it assumes a convex surface; and when a quantity of it is thrown on the floor, it is dispersed into a variety of little globules, which can scarcely be separated from one another. Even mercury, the most perfect of all the fluids, is endowed with such a cohesive force among its particles, that if a glass tube, with a small bore, is immersed in a vessel full of this fluid, the mercury will be lower in the tube than the surface of the surrounding fluid—if a small quantity of it be put in a glass vessel, with a gentle rising in the middle of its bottom, the mercury will desert the middle, and form itself into a ring, considerably rounded at the edges; or if several drops of mercury be placed upon a piece of flat glass, they will assume a spherical form; and if brought within certain limits, they will conglomerate and form a single drop. Now, all these phenomena concur to prove, that the particles of water have a mutual attraction for each other; that the particles of mercury have a greater attraction for one another, than for the particles of glass; and, consequently, that these substances are not entitled to the appellation of perfect fluids.

44. It was universally believed, till within the last seventy years, that water, mercury, and other fluids of a similar kind, could not be made to occupy a smaller space, by the application of any external force. This opinion was founded on an experiment made by Lord Bacon, who inclosed a quantity of water in a leaden globe, and by applying a great force attempted to compress the water into a less space than it occupied at first: The water, however, made its way through the pores of the metal, and stood on its surface like dew. The same experiment was afterwards repeated at Florence by the Academy del Cimento, who filled a silver globe with water, and hammered it with such force as to alter its form, and drive the water through the pores of the metal. Though these experiments were generally reckoned decisive proofs of incompressibility, yet Bacon himself seems to have drawn from his experiment a very different conclusion; for after giving an account of it, he immediately adds, that he computed into how much less space the water was driven by this violent pressure. This Hydrostatic passage from Lord Bacon does not seem to have been noticed by any writer on hydrostatics, and appears a complete proof that the compressibility of water was fairly deducible from the issue of his experiment. In consequence of the reliance which was universally placed on the result of the Florentine experiment, fluids have generally been divided into compressible and incompressible, or elastic and non-elastic fluids: water, oil, alcohol, and mercury, being regarded as incompressible and non-elastic; and air, steam, and other aeriform fluids, as compressible or elastic.

45. About the year 1761, the ingenious Mr Canton began to consider this subject with attention, and distrusting the result obtained by the Academy del Cimento, resolved to bring the question to a decisive issue. Having procured a small glass tube, about two feet long, with a ball at one end, an inch and a quarter in diameter, he filled the ball and part of the tube with mercury, and brought it to the temperature of 50° of Fahrenheit. The mercury then stood six inches and a half above the ball; but after it had been raised to the top of the tube by heat, and the tube sealed hermetically, then, upon bringing the mercury to its former temperature of 50°, it stood 1/18th of an inch higher in the tube than it did before. By repeating the same experiment with water exhausted of air, instead of mercury, the water stood 45/18th of an inch higher in the tube than it did at first. Hence, it is evident, that when the weight of the atmosphere was removed, the water and mercury expanded, and that the water expanded 1/18th of an inch more than the mercury. By placing the apparatus in the receiver of a condensing engine, and condensing the air in the receiver, he increased the pressure upon the water, and found that it descended in the tube. Having thus ascertained the fact, that water and mercury are compressible, he subjected other fluids to similar experiments, and obtained the results in the following table:

| Compression of Mercury | Sea-Water | Rain-Water | Oil of Olives | Spirit of Wine | |------------------------|-----------|------------|--------------|---------------| | 3 | 13.595 | 1.028 | 0.918 | 0.846 |

Lest it should be imagined that this small degree of compressibility arose from air imprisoned in the water, Mr Canton made the experiment on some water which had imbibed a considerable quantity of air, and found that its compressibility was not in the least augmented. By inspecting the preceding table, it will be seen that the compressibility of the different fluids is nearly in the inverse ratio of their specific gravities.

46. The experiments of Mr Canton have been since confirmed by Professor Zimmerman. He found that sea-water was compressed 1/18th part of its bulk, when inclosed in the cavity of a strong iron cylinder, and under the influence of a force equal to a column of sea-water 1000 feet in height. From these facts, it is obvious that fluids are susceptible of contraction and dilatation, and that there is no foundation in nature for their being divided into compressible and incompressible. If fluids are compressible, they will also be elastic; for when the compressing force is removed, they will recover their former magnitude; and

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1 Bacon's Works, by Shaw, vol. ii. p. 521; Nouum Organum, part. ii. sect. 2. alph. 45. § 222. 2 See the Philosophical Transactions for 1762 and 1764, vols. iii. and iv. hence their division into elastic and non-elastic is equally improper.

A series of very valuable experiments on the compressibility of water has since that time been made by Professor Oersted of Copenhagen. At the temperature at which water has a maximum density, which, according to Professor Stamper of Vienna, is at 38° 75' Fahrenheit, Professor Oersted found the true compressibility of water by one atmosphere (or 336 French lines of mercury) to be 46.1 millionths of the volume; the difference between the true and the apparent compressibility arising from the effect of the heat developed by the compression, by which the liquid and the bottle are dilated. He found also that the differences of volume in the compressed water are proportional to the compressing power; and that this law holds as far as the pressure of 65 atmospheres, and probably much farther; but how far he was not able to determine, as his apparatus could not resist a greater pressure. He repeated Canton's experiments with great care, and verified his results. Oersted confirms Canton's result that water is less compressible during summer than during winter. (See Report of Third Meeting of British Association.)

Mr Perkins proved the compressibility of water by subjecting that fluid to mechanical pressure; for a description of his apparatus, see Philosophical Transactions for 1820.

Colladon and Storm have made experiments on the compressibility of different liquids, and under a pressure of one atmosphere they assign the following:—Water, 0.0000496; alcohol, 0.0000916; mercury, 0.00000338, each of its own volume.

CHAPTER I.—ON THE PRESSURE AND EQUILIBRIUM OF FLUIDS.

48. Prop. I.—When a pressure acts on the surface of a perfect fluid which is without weight, and at rest, it is equally pressed in every direction.

As it is the distinguishing property of fluids that their particles yield to the smallest pressure, and are easily moved among themselves (26), it necessarily follows, that if any particle is more pressed towards one side than towards another, it will move to that side where the pressure is least; and the equilibrium of the fluid mass will be instantly destroyed. But by the hypothesis the fluid is in equilibrium, consequently the particle cannot move towards one side, and must therefore be equally pressed in every direction.

In order to illustrate this general law, let EF (fig. 1) be a vessel full of any liquid, and let mn, op be two orifices at equal depths below its surface; then, in order to prevent the water from escaping, it will be necessary to apply two pistons, A and B, to the orifices mn, op with the same force, whether the orifice be horizontal or vertical, or in any degree inclined to the horizon; so that the pressure to which the fluid mass is subject, which in this case is its own gravity, must be distributed in every direction. But if the fluid has no weight, then the pressure exerted against the fluid at the orifice op, by means of the piston B, will propagate itself through every part of the circular vessel EF, so that if the orifices mn, tu are shut, and rs open, the fluid would rush through this aperture, in the same manner as it would rush through mn or tu, were all the other orifices shut. This proposition, however, is true only in the case of perfect fluids; for when there is a sensible cohesion between the particles, as

These results are somewhat different from those of Oersted, but this may be owing to the compression of the tubes or vessels.

M. Regnault made (in 1854) very accurate experiments on the compressibility of fluids with an apparatus a little different from that which Oersted used; he found the following results:—The volume of a mass of water subjected to pressure diminishes by 0.000048 of its volume for every atmosphere; i.e., if a mass of water having a volume of a million litres, when its surface is subjected to no compressing force, comes to be subjected to a pressure of 1, 2, 3, 4, &c., atmospheres, its volume will diminish by 48 litres, twice 48 litres, thrice 48 litres, four times 48 litres, &c., respectively. When mercury is the fluid pressed, then it diminishes by 0.0000035 of its volume for every atmosphere incumbent on its surface.

47. The doctrines of hydrostatics have been deduced by different philosophers from different properties of fluids. Euler has founded his analysis on the following property, "that when fluids are subjected to any pressure, that pressure is so diffused throughout the mass, that when it remains in equilibrium all its parts are equally pressed in every direction." D'Alembert at first deduced the principles of hydrostatics from the property which fluids have of rising to the same altitude in any number of communicating vessels; but he afterwards adopted the same property as Euler, from the foundation which it furnishes for an algebraical calculus. The same property has been employed by Bossuet, Prony, and other writers, and will form the first proposition of the following chapter.

49. Prop. II.—If to the equal orifices mn, tu, op, rs, of Fig. 1, a vessel, containing a fluid destitute of weight, be applied equal powers A, B, C, D, in a perpendicular direction, or if the orifices mn, &c., be unequal, and the powers A, B, &c., which are respectively applied to them be proportional to the orifices, these powers will be in equilibrium.

It is evident from the last proposition, that the pressure exerted by the power B is transmitted equally to the orifices mn, rs, tu, that the pressure of the power C is transmitted equally to the orifices mn, op, tu, and so on with all other powers. Every orifice, then, is influenced with the same pressure, and, consequently, none of the powers A, B, C, D can yield to the action of the rest. The fluid mass, therefore, will neither change its form nor its situation, and the powers A, B, C, D will be in equilibrium. If the powers A, B, C, D are not equal to one another, nor the orifices mn, op, rs, tu; but if A : B = mn : op, and so on with the rest, the fluid will still be in equilibrium. Let A be greater than B, then mn will be greater than op; and whatever number of times B is contained in A, so many times will op be contained in mn. If A = 2B, then mn = 2op, and since the orifice mn is double of op, the pressure upon it must also be double; and, in order to resist that pressure, the power A must also be double of B; but, by hypothesis, A = 2B, consequently the pressures upon the orifices, or the powers A, B, will be in equilibrium. If the power A is any other multiple of B, it may be shown in the same way that the fluid will be in equilibrium.

The application of the principle contained in this propo-

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1 Mem. de l'Inst., 1854. 2 Nov. Comment. Petropoli, tom. xlii, p. 305. 3 Mélanges de Littérature, d'Histoire, et Philosophie. 4 Traité des Fluides, § 20. 50. Prop. III.—The surface of a fluid of small extent at rest, and subject to gravity, is horizontal.

The truth of the proposition is a necessary consequence of gravity acting on the fluid particles. For the attraction of gravity being supposed to be lodged in the centre of the earth, and a surface of small extent being incomparable with respect to the distance of the attracting centre, we may suppose, without sensible error, that the particles on this surface of small extent are all pulled with the same intensity, and, therefore, they must all be in the same plane, which will be horizontal. For if the plane be not horizontal, let fig. 2 be a section of the mass, with its surface BSA; take A and B, particles of its surface, and BC, AD, vertical columns of liquid molecules, C and D being in the same horizontal line. Now, neglecting the atmospheric column, the pressures at A and B are nothing, but at D and C they are equivalent to the weight of two liquid cylinders AD and BC, having equal bases. Hence, since there is a difference of weights in the columns, A and B cannot be on the same level, and, therefore, the surface BSA of the fluid cannot be in equilibrium. Again, a particle as S would seek its lowest level, and so all others similarly situated, until the whole surface particles over a small extent would be in the same plane; and the pressure at all points of the surface of a fluid being the same, the fluid is in equilibrium, and therefore a fluid surface of small extent will be horizontal. The principle of the proposition is applied in the art of levelling.

51. As the direction of gravity is in lines which meet near the centre of the earth; and as it appears from this proposition, that the surface of fluids is perpendicular to that direction, their surface will be a portion of a spheroid similar to the earth. When the surface has no great extent, it may be safely considered as a plane; but when it is pretty large, the curvature of the earth must be taken into account.

52. Prop. IV.—Fluids rise to the same level throughout a system of communicating vessels.

Suppose that there were alongside of each other several vessels, A, C, E, G, I (fig. 3), of different forms and capacities, cylindrical, spiral, conical, or spherical, containing fluid, and communicating with one another by means of a common vessel P. Now, if water be poured into this vessel, it follows from Prop. III. (50), that the fluid will stand at a common level in each of the vessels A, C, E, G, I; and if a line be drawn parallel to one surface, it will be parallel to all the surfaces, and it will also be parallel to the horizontal plane. Therefore, also, if we remove the different shaped vessels from the common reservoir, and introduce into the vessel ABEF solid bodies, T, GFH, UK, &c., then, after each immersion, the surface will still be horizontal; and when all the solids have been immersed, we shall have, as it were, a system of communicating vessels in which the surface of the fluid will necessarily be horizontal.

53. When the communicating vessels are so small, that they may be regarded as capillary tubes, the surface of the fluid will not be horizontal, as will afterwards be shown. (See Capillary Attraction, art. 114.)

54. This proposition explains the reason why the surface of small pools in the vicinity of rivers is always on a level with the surface of the rivers themselves, when there is any subterraneous communication between the river and the pool. The river and the pool may be considered as communicating vessels.

55. The practical application of the preceding proposition is finely shown in the methods employed for supplying towns with water. The ancients built aqueducts for this purpose, perhaps not so much from ignorance of the principle that water always seeks its own level, as from the difficulty of making their pipes water-tight at the joints. In modern times, pipes carefully jointed and secured are almost invariably used. The New River Water-works, however, which supply part of London with water, embrace both aqueducts and pipes.

56. Prop. V.—The pressure at any point below the surface of a fluid subject to gravity and at rest, varies as its depth.

Let point B (fig. 4) be immediately under the surface at a depth AB. The series of particles in the column AB is at rest, and may be supposed separated from the rest of the fluid mass, and lodged in a tube, not capillary, or the whole may be supposed rigid or solid like ice. Now, every particle of the column has weight, and so the upper will press upon those lower, and these again on the lowest particles; hence, the pressure on the point B will be equal to the aggregate weight of all the liquid particles; and hence, also, the pressure will vary as the depth of the point below the surface.

Similarly, also, let the point subject to pressure be C (fig. 5), not directly under the surface. Take ED a vertical from the surface, and draw DC horizontal; then we may consider the series of particles DC as a solid mass. Now, as the pressure is the same at all points along the line DC, and as the pressure at D is equivalent to a weight of fluid particles DE, so also is the pressure at C of the same value. Therefore, DE being equal to FC, the pressure at C varies as its depth below the surface of the fluid.

57. Prop. V.—To find the pressure on a line sunk in a fluid subject to gravity.

Let the line be parallel to the surface of the fluid, and the portion of it the pressure on which is required be the unit of length. Let MN (fig. 4) be this portion of the given line at the depth AM. There may be now supposed to be a solidified mass of particles, with base MN and height AM, resting on MN, each particle of which has weight, and wholly separate from the rest of the mass. If, then, a unit of bulk weigh \( g_p \), where \( g \) expresses the action of gravity on one particle, and \( p \) the density of the fluid; and if \( z \) be the depth of the line, or the number of units in the depth over the unit of length, the weight of the whole column will be expressed by \( g \cdot z \). But this weight is the same as the pressure to which the unit of length is exposed; let this be \( p \), therefore \( p = g \cdot z \).

The same will be the result, if the portion MN were lying oblique to the horizontal surface, the mean depth being \( z \).

58. Cor.—Hence, if we have A units of length, or a portion of a line of which the length is A, then if its depth be \( z \), the pressure will be \( p \cdot A = g \cdot z \cdot A = P \).

The quantity \( p \) is the same for the same fluid, but different for different fluids. Further, in problems of a practical nature, the pressure of the atmosphere on any line or area, equal to that sunk in the fluid, must be added on to the weight of the fluid, so as to obtain the true pressure.

59. Prop. VI.—If a surface be sunk in a fluid subject to gravity, find the pressure upon it, or on the surface of the vessel containing fluid.

Let the given area be A, and suppose it broken up into an infinity of small areas, \( a_1, a_2, a_3, \ldots, a_n \), the mean depths of which under the surface let be \( z_1, z_2, z_3, \ldots, z_n \); then the different pressures on these partial areas will be (Prop. V.) respectively \( g \cdot z_1 \cdot a_1, g \cdot z_2 \cdot a_2, g \cdot z_3 \cdot a_3, \ldots, g \cdot z_n \cdot a_n \); therefore, the total pressure on the surface is \( g \cdot z_1 \cdot a_1 + g \cdot z_2 \cdot a_2 + g \cdot z_3 \cdot a_3 + \ldots + g \cdot z_n \cdot a_n \). But if Z be the depth of the centre of gravity below the surface, and S the area of the plane, then, by a property of the centre of gravity (see Mechanics), we have \( g \cdot Z \cdot S = g \cdot z_1 \cdot a_1 + g \cdot z_2 \cdot a_2 + g \cdot z_3 \cdot a_3 + \ldots + g \cdot z_n \cdot a_n \). But \( g \cdot Z \cdot S \) is evidently the weight of a column of fluid, of which the height is Z and base is S.

Therefore, the pressure of a fluid on any surface is the weight of a column of the fluid, the base of which is the area of the surface pressed, and the height, the depth of the centre of gravity of the area below the surface of the fluid.

60. Cor. I.—From this proposition it follows, that the whole pressure on the sides of a vessel which are perpendicular to its base, is equal to the weight of a rectangular prism of the fluid, whose altitude is that of the fluid, and whose base is a parallelogram, one side of which is equal to the altitude of the fluid, and the other to half the perimeter of the vessel.

Cor. 2.—The pressure on the surface of a hemispherical vessel full of fluid, is equal to the product of its surface multiplied by its radius.

Cor. 3.—In a cubical vessel the pressure against one side is equal to half the pressure against the bottom; and the pressure against the sides and bottom together, is to that against the bottom alone, as three to one. Hence, as the pressure against the bottom is equal to the weight of the fluid in the vessel, the pressure against both the sides and bottom will be equal to three times that weight.

Cor. 4.—The pressure sustained by different parts of the side of a vessel are as the squares of their depths below the surface; and if these depths are made the abscissae of a parabola, its ordinates will indicate the corresponding pressures.

61. Prop. VII.—To find the pressure on the bottom of a vessel of any form containing fluid.

If the walls of the vessel be vertical as \( \alpha \) (fig. 6), and its base parallel to the surface, it is clear that the pressure on the bottom will be equal to the whole weight of fluid contained in \( \alpha \). But if the containing vessels be \( \beta \) and \( \gamma \) of fig. 6, then the pressure on the base AB in the former case is greater than the weight of the fluid, while the latter is less than that weight. For the pressure on AB of \( \beta \), is the weight of a column ABDC, which is heavier than the contained fluid, and the pressure on AB of \( \gamma \) is a column ABDC, which is lighter than the contained fluid.

If the vessel be as \( \beta \), then a pressure, \( P \), arising from the resistance of the side at a point \( o \), will act perpendicularly inward, with a force proportioned to the depth of \( o \); and the same will be the case for the several pressures at the different points of the inclining sides; but these pressures, as \( P \), may each be resolved into vertical, as \( S \), and horizontal, as \( R \), components, of which all the horizontal ones for every section balance, whereas the vertical components acting downwards and parallel will have a single resultant, which will be an amount of additional fluid, as it were, pressing on the base. On the other hand, in \( \gamma \), of fig. 6, the pressures, as \( P \) in points, as \( o \), of the sides acting perpendicularly outwards, may be resolved into vertical and horizontal components; the latter destroy each other at every section; the former, acting upwards or downwards, according to the position of \( o \), have a single resultant, acting in opposition to the general weight of the fluid, which is the same as if a quantity of fluid were removed from the vessel. Hence, if the bases be all equal, and the fluid at the same height in each vessel, whatever be their forms, the pressure on base will be equal to the weight of a column of fluid, having for its base that of the vessel, and for its height the depth of the base below the surface of the fluid.

62. In order to illustrate the preceding propositions, let there be four vessels, A, B, C, D (fig. 7), having bottoms all of the same area, and closed by plates E, F, G, H, of the same weight. Let the plates also be kept in their places by means of strings passing over pulleys, and acted on by the equal weights W, W', W", W". These weights will measure the pressure of water on the plates, i.e., on the bases of the vessels. For if water be poured into each vessel till there be just enough to balance the weight, or when the plates are about to descend, it will be found that the water will be at the same height in each vessel. The same will be the case, whatever be the shape of the vessels, even although one may be a thousand times more capacious than another.

63. Definition.—The centre of pressure is that point of a surface exposed to the pressure of a fluid, to which, if the total pressure were applied, the effect upon the plane would be the same as when the pressure was distributed over the whole surface: Or, it is that point to which, if a force equal to the total pressure were applied in a contrary direction, the one would exactly balance the other or, in other words, 64. Prop. VIII.—To find the centre of pressure of any plane surface immersed in a fluid to any depth.

Let \(ABC\) (fig. 8) be a plane surface sunk in a fluid, and inclined at an angle, \(\theta\), to the surface of the fluid which it meets, when produced, in HI. Let \(l = AG\) be the distance of point A from the water-line, or HI, measured along \(GAx\) drawn at right angles to HI in the plane of \(ABC\); let A be the origin of rectangular co-ordinates to which \(ABC\) is referred, \(GAx\) the axis of \(x\), and \(AY\), parallel to HI, that of \(y\). EF is an elemental area of the surface \(ABC\).

Take \(Y\) and \(X\) as the rectangular co-ordinates of the centre of pressure.

Now the pressure at every point equals the height of a column of particles, and this being so, we may take all the pressures on \(ABC\) as parallel forces. Hence, from Statics, we have,

\[X\text{pressure on } ABC = \text{moment of pressure on } ABC(1)\text{, and}\] \[Y\text{pressure on } ABC = \text{moment of pressure on } ABC(2).\]

If we suppose area \(ABC\) to be broken up into a series of very small surfaces as EF, of which \(dx\) and \(dy\) are the dimensions, it is evident that the pressure on the whole surface \(ABC\) will be the sum of the pressures on these small elements. But the pressure on the area \(EF = p \cdot dx \cdot dy = gp \times \text{the perpendicular from } EF \text{ to } HI \cdot dx \cdot dy = gp(x + l) \sin \theta \cdot dx \cdot dy\); where \(x\) is the absciss of the infinitely small area \(EF\):

\[\therefore \text{The whole pressure on } ABC = \iint p \cdot dx \cdot dy = gp \cdot \sin \theta \iint (x + l) \cdot dx \cdot dy.\]

If we multiply this last by \(x\), it will give us the moment about \(AY\), and if by \(y\), it will give us the moment about \(AX\). Therefore, moment of pressure on \(ABC\) about \(AY\)

\[= gp \sin \theta \iint (x + l) \cdot x \cdot dx \cdot dy;\]

and, moment of pressure on \(ABC\) about \(AX\)

\[= gp \sin \theta \iint (x + l) \cdot y \cdot dx \cdot dy.\]

Substitute now in (1) and (2) the values thus found,

\[\therefore X \cdot gp \sin \theta \iint (x + l) \cdot dx \cdot dy = gp \sin \theta \iint (x + l) \cdot x \cdot dx \cdot dy;\] \[Y \cdot gp \sin \theta \iint (x + l) \cdot dx \cdot dy = gp \sin \theta \iint (x + l) \cdot y \cdot dx \cdot dy;\]

or, \(X \iint (x + l) \cdot dx \cdot dy = \iint (x + l) \cdot x \cdot dx \cdot dy(3);\) \[Y \iint (x + l) \cdot dx \cdot dy = \iint (x + l) \cdot y \cdot dx \cdot dy(4).\]

Since equations (3) and (4) will give a value for \(X\) and \(Y\) respectively, the centre of pressure is known. If the surface be curved, and its curvature known, then, on integrating, we have,

\[X \iint (x + l) \cdot y \cdot dx = \iint (x + l) \cdot x \cdot y \cdot dx\] \[Y \iint (x + l) \cdot y \cdot dx = \iint (x + l) \cdot y^2 \cdot dx,\]

which are very important formulae.

If any point of the surface \(ABC\) coincide with the surface, then \(l = 0\);

\[\therefore X \iint x \cdot y \cdot dx = \iint x^2 \cdot y \cdot dx \text{ and } Y \iint x \cdot y \cdot dx = \iint x^2 \cdot y^2 \cdot dx.\]

If the surface be sunk to a great depth, or \(l = \infty\), then will the centre of gravity coincide with the centre of pressure; for, from (3) and (4), since the infinite parts must be equal,

\[X \iint dx \cdot dy = \iint x \cdot dx \cdot dy; \text{ or, } X \iint y \cdot dx = \iint x \cdot y \cdot dx;\] \[Y \iint dx \cdot dy = \iint y \cdot dx \cdot dy; \text{ or, } Y \iint y \cdot dx = \iint y^2 \cdot dx.\]

But the values of \(X\) and \(Y\), determined from these last equations, are the co-ordinates of the centre of gravity of a plane area bounded by a plane curve. Hence, under such a condition, the centre of pressure is the same as the centre of gravity.

65. Cor. 1.—A physical plane, one side of which is exposed to the pressure of a fluid, may be kept at rest by a single force, equal and opposite to the pressure of the fluid applied at its centre of pressure.

Cor. 2.—Where a plane lamina of a very small uniform thickness is moveable round its \(Y\) axis, the values of \(X\) and \(Y\) will be those of the co-ordinates of its centre of percussion.

Cor. 3.—The centre of pressure of a triangle, the base of which coincides with the level of the water, is at a distance of one-third the height of the triangle from its base. If the summit coincide with the surface, then the centre of pressure will be at two-thirds the height, measuring from the summit.

66. The application of the preceding proposition is of the utmost consequence in the case of flood-gates, sea-walls, dykes, brewers' vats, and the like.

SECTION II.—INSTRUMENTS FOR ILLUSTRATING THE PRESSURE OF FLUIDS.

67. We have already seen from arts. 59, 61, 62, that the pressure on the bottom of a vessel of any form, does not depend on the quantity of fluid contained therein, but on the hydrostatic area of the base and the depth of the fluid. The hydrostatic bellows is an excellent instrument for showing this principle. A and B (fig. 9) are two strong boards, united together by means of leather, so as to form a sort of bellows, but perfectly water-tight. C is a vertical tube communicating with the interior of the bellows. If water, or any other fluid, be poured down the tube C by the funnel F, it will enter the bellows AB, and raise the upper board A, even though loaded with a heavy weight W. The upper board A will continue to be forced up till either the bellows be full or the tube be filled throughout its whole length. Suppose that an equilibrium subsists between the water in the tube and bellows, and let DE be the height of the column of water in the tube above the surface of water in the bellows. Then the weight of this column DE will be that which supports the whole weight raised; for the fluid DC in the tube is in equilibrium with that in the bellows, and hence the pressure at the surface of the fluid in the bellows is in equilibrium with the column DE. 68. Suppose that the whole weight raised is $W$, which includes the load, the weight of the upper board and of the leather, let $S$ be the area of the upper board, $A_s$ the sectional area of the tube; let also $p$ be the unit of pressure on a unit of surface transmitted from the surface at $D$ by the column $DE$; then, since fluids press equally in every direction, the upward pressure on the board $A$ will = $p \cdot S$, which balances the downward pressure of the weight $W$. Hence $p \cdot S = W$. Now, the whole pressure at $D = p \cdot s$; therefore we have $W = p \cdot S$, and pressure at $D = p \cdot s$.

$$\therefore W : \text{pressure at } D = p \cdot S : p \cdot s = S : s.$$

$$\therefore W = \text{weight raised} = \frac{S}{s} : \text{pressure at } D,$$

$$= \frac{S}{s} : \text{weight of a column } DE.$$

Hence the weight which can be raised will be increased if we increase the area of the upper board, or if we diminish the sectional area of the tube, or if we increase the pressure of the column, i.e., if we lengthen the tube.

69. In a theoretical point of view the ratio $\frac{S}{s}$ may become infinitely large; hence any quantity of fluid, however small, may be so employed as to sustain any weight however large.

The same instrument may be employed for the same purpose when air or any gas is blown or forced in at the funnel $F$, and condensed in the bellows. In which case a very heavy weight may be raised by means of a small vertical tube. Hence, a person standing on the hydrostatic bellows might raise himself by blowing into the tube.

Hence, also, a small vertical pipe like that in fig. 9, fitted into a barrel full of water or other fluid, may, when filled with water or other fluid, be a means of bursting the barrel.

Similarly, also, it is highly probable, that the sudden formation of inland lakes is due to this pressure of water. For, suppose that a considerable extent of water-tight stratum forms the roof of a subterranean lake, and that this water, and therefore the roof, is subject to the pressure of a column several hundred feet high; it is clear that, as soon as the upward pressure of the column exceeds the strength of the material forming the roof, the latter will break up and a lake will be formed.

70. The siphon is a bent tube, $ABCDE$ (fig. 10), having two legs, one shorter than the other, wherewith when the shorter end is plunged into a vessel containing fluid, it may be emptied into another without there being any communication between the vessels. Before a transference of the liquid takes place the tube is previously filled with fluid, and its ends stopped by the fingers, then inverted, and the shorter end placed into the vessel which is to be emptied; the other end being left free, the water escapes by it, and will continue to escape until the level of the water in the vessel be in the same plane as the mouth of the short leg.

71. In order to explain the flow of water through the tube, let us suppose that the end $A$, after the siphon has been filled, is made water-tight by means of a moveable piston, and that the surface of $M$ is open to the atmosphere. The inner and outer surfaces of the piston will be exposed to pressure, so also will the surface of fluid in $M$; the latter, as also the fluid in the short leg, will be subject to the atmospheric pressure, but the former will have the pressure on the outer face constant, while that on the inner face will depend on its position in the long leg. As the piston rises from $A$ in the siphon leg, the pressure on the inner face of the piston gradually diminishes. If the piston be at $F$, or in the horizontal part $C$, its outer face will be subject to the atmospheric pressure, but its inner face will be subject to the pressure exercised on the fluid surface of $M$, less the weight of a column of fluid, the height of which is the vertical distance of the inner face of the piston above the surface of the fluid in $M$. Let the piston be gradually drawn downwards, and the pressure on the inner face increases, since, on arriving at $B$, which is on the same level as the fluid surface in $M$, it equals that of the atmosphere.

Hence, if the siphon were of the form $BFCD$, and $B$ and $D$ in the same horizontal line, no water would flow out at $B$, simply because the pressure at $B$ is the same as that at $D$. Let, now, the piston be lower than $B$, then its inner face pressed not only by the atmosphere on the fluid surface of $M$, but also by the weight of a fluid column of a height equal to the distance between it and $B$. The water will then flow out by this leg. The lower, therefore, the piston is, the greater will be the excess of the pressure on the inner over that on the outer face; and when it comes to the mouth of the tube at $A$ this excess will be a maximum; hence, as soon as the piston has been withdrawn, a flow of water will take place by the mouth $A$, with a velocity equal to that with which a body, falling freely from $B$, would have on reaching $A$. The velocity of escape, however, will not be accurately the above, owing to the bending of the tube, and also to the friction of the water in passing along the siphon.

72. The pressure of the atmosphere is then a very important matter in the action of the siphon, since without this element the fluid would not rise in the shorter leg. If a prepared siphon, i.e., filled with fluid, and its ends temporarily stopped, were set to act under the exhausted receiver, the fluid would divide itself into two portions; that in the long leg would fall out on one side, while that in the short leg would fall back into the fluid in the vessel. Again, if the siphon should have, between the surface of the fluid to be emptied and its bend, a distance of more than 32 feet, then it will not work; if nearly 32 feet, its action will be very sluggish.

73. The filling of the siphon with fluid so as to prepare it for working, is not the best method; generally a small tube rises vertically near $A$, as shown by the dotted lines (fig. 10). The air is drawn out of the siphon by means of this small tube, and as the air is more and more rarified, the water rises to the top of the short leg, then flowing along the horizontal portion, if any, enters the long leg and passes out. It is in this way that tuns of wine and other spirits are emptied, without requiring to pierce a hole in the lower end, the bung hole being open; it will suffice if a white-iron siphon be introduced into the bung hole, and the space around the plunged leg be free that the pressure of the atmosphere may be allowed to act.

74. The application of the principle of the siphon in nature is finely shown in the action of reciprocating, or intermittent springs, that is, of those which flow and cease at regular or irregular periods. A large subterraneous cavern, or porous bed, may be regarded as a reservoir collecting water by narrow pipes or channels; and the only outlet to this collected water will be by natural or artificial pipes rising to the surface, forming, in every respect, a natural siphon. The reservoir will evidently be drained by the latter pipes, whenever a sufficient head or pressure is upon the collected water. Hence, also, Artesian Wells are but natural siphons. CHAPTER II.—ON FLOTATION AND SPECIFIC GRAVITIES.

SECTION I.—FLOTATION.

75. The consideration of floating bodies enables us to determine the conditions of rest or motion when placed in a fluid; if at rest, whether their position will be stable or unstable, and if in motion, the circumstances under which they rise or sink in the fluid.

76. Prop. I.—A body at rest and immersed in a fluid, is forced upwards by a pressure equal to the weight of the fluid displaced.

Let us suppose first of all, that we have in the vessel of fluid (fig. 11) a cubical mass M. When the surface of M is horizontal, the fluid is at rest, and so also is M. The vertical faces of M are pressed equally, and the forces are balanced among themselves; but the lower and upper face sustain different pressures; the excess lies on the former, and is equal to the weight of a cubical mass, M, of fluid. Now, since M, which we may suppose to be a mass of the fluid solidified, is at rest, its weight will act downwards, and the supporting fluid will exert, as it were, a pressure forcing it upwards; and as M is in equilibrium, the downward and upward pressures will be equal to each other. Suppose now that we remove the fluid mass M, and introduce a body of a different material, such that the equilibrium shall still be unchanged, it is evident that we shall have the same upward force of the supporting fluid, and downward force of the body, balancing each other. If the weight of the body immersed be equal to that of the fluid displaced, as will then be the case, the body will be at rest; if the former weight should exceed the latter the body will sink, if the latter the former, the body will float on the surface.

77. Again, the body immersed and at rest in the fluid may be supposed to be of any form as N (fig. 11); then it will be supported in the same way as a similarly conditioned mass of the fluid would be, that is, the upward and downward pressure of the supporting fluid and body respectively will be the same. Hence the truth of the proposition.

This proposition is known as the principle of Archimedes, having been first discovered by him.

78. Definition.—When a body floats in a fluid, the plane passing through the body, and coinciding with the surface plane of the fluid, is called the plane of flotation.

79. Prop. II.—When a body floats in a fluid, the weight of the volume which it displaces is equal to the weight of the body, and the downward force of the body, as also the upward force of the fluid, is equal to the weight of fluid displaced.

Let S be a body floating in the vessel (fig. 11), a mass M' being above, and M' being below the plane of flotation, so that the whole mass, say M = M' + M'. Now, evidently, when S floats, it is buoyed up by some under pressure, and before immersion took place, a volume of fluid was sustained in the same manner; as we have then an upward pressure supporting at one time a volume M' of fluid, and again a body, a mass of which M' is immersed, it follows that the weight of the floating body and that of the volume of fluid displaced are equal to each other.

Again, if we take any point of the surface of the mass below the plane of flotation, then the pressure at this point of the body will be the weight of a fluid column of a height equal to the depth of that point below the surface; and the same holds true for all the other pressures on the other points of the mass immersed, i.e., the weights of the several fluid columns. But the weight of all the fluid columns equals that of the fluid displaced, hence the downward pressure of the body and the upward pressure of the sustaining fluid each equals the weight of the fluid displaced.

Call V the volume of the floating body, ρ its density, then its weight is (M + M') g = Vρg. Call V' the volume of M', and ρ' the density of the fluid, then weight of displaced fluid = V'ρ'g; ∴ Vρg = V'ρ'g, or Vρ = V'ρ'. If ρ = ρ', then V = V', and the body will be wholly under the fluid.

80. Cor.—From this proposition, we learn that a floating body is buoyed up by a force equal to the weight of the fluid displaced. This follows from the principle of Archimedes, and means that a body either floating or wholly immersed, loses as much of its weight as equals that of the fluid displaced. Hence a bucket full of water placed in its own medium is lighter than when surrounded by air.

The application of the preceding proposition is finely shown in Green's Canal Lifts, whereby laden barges may be transferred vertically from one level to another of a canal; for an account of which we refer to the Trans. Inst. Civ. Engin. 4to, vol. ii., p. 185.

81. When a body is not at rest in a fluid, it will ascend or descend with a moving force equal to the difference between the weight of the solid, or part of the solid immersed, and the weight of an equal volume of the fluid. It is on this principle that the Camel is explained. Two large boxes or chests, with a number of separate compartments in each, are filled with water, and sunk one on each side of a sunken vessel which is to be raised, the chests being made fast to the keel by straps passing underneath it. The water is now pumped out, and the buoyant power of the chests raises the vessel. This is also the principle on which life-preservers act. Fish also owe their equilibrium to the same principle, since on rising or sinking the weight of the column of water diminishes or increases and so expands or contracts their body.

82. Definition—Metacentre.—Let the body be slightly disturbed through a small angle, then by the metacentre is meant, the point of intersection of the vertical line through the centre of gravity of the fluid displaced, and the vertical line through the centre of gravity of the body when at rest.

83. Prop. III.—If a body float in a fluid, the line joining the centres of gravity of the body, and the fluid displaced is in the same vertical.

The floating body, we have seen, presses downwards, and the fluid presses upwards with an equal force; we may therefore reckon these pressures as forces acting respectively at the centre of gravity of the body, downwards, and at the centre of gravity of the fluid displaced, upwards. Hence, the body cannot be at rest, unless these forces be equal and opposite, i.e., unless the line joining the centres of gravity of the body and of the fluid displaced be in the same vertical.

84. Cor.—In all other positions the forces will constitute what is called a statical couple, which admits of no single resultant of the two components. 85. We can now determine the conditions under which a floating body will remain at rest: these are—

1st, That the weight of the body be equal to that of the fluid displaced; and

2nd, That the centre of gravity of the body, and the centre of gravity of the fluid displaced, be in the same vertical.

86. In all cases of the equilibrium of floating bodies these conditions must obtain. They are insufficient, however, to determine the nature of the equilibrium, but refer to a motion of translation upwards, downwards, or forwards, not to a motion of rotation. Thus, suppose a body floating in a fluid to be at rest, and let it be struck obliquely, then the body will move under the influence of two motions—one of translation, the other of rotation. By the first, the body will oscillate backwards and forwards, till, after a time, it will resume its original position. By the second, the motion will differ with the nature of the floating body; but these motions may be referred to three classes:—1st, When the body oscillates about its original position, and at last returns to it. 2nd, When, after passing through a certain angle, the body remains at rest, without recovering its first position. 3rd, When it has no tendency whatever to come back to its first position, but continues to recede the farther from it.

87. In these three cases we have the nature of the equilibrium of floating bodies. Hence the first gives us an equilibrium of stability, the second one of neutrality or indifference, and the third, one of instability.

A floating body, therefore, is said to be in stable equilibrium, when, on suffering a slight displacement, or being made to revolve through a certain angle, it returns to its first position.

A floating body is said to be in neutral or indifferent equilibrium, when, on being slightly disturbed, it will rest in any position.

A floating body is said to be in unstable equilibrium, when, on receiving a slight disturbance, it continually departs from its first position.

88. The nature of the equilibrium of a floating body is a very important matter, more especially in ship-building, and in the distribution of the cargo, as also of the ballast of a vessel.

89. Prop. IV.—To determine whether the equilibrium is stable, unstable, or neutral.

Let A B (fig. 12) be a body floating in a fluid, g the centre of gravity of the body, mg the direction of gravity when the body was at rest, as in the dotted figure, but which, when the body has been displaced, lies in gm; g' the centre of gravity of the fluid displaced. Pass a vertical through g', meeting gm in m; m is called the metacentre of the body. Now, the weight of the body at g acting down, and the pressure of the fluid at g' acting up, tend to bring the body back to its position of rest. Hence, when m is above g, the equilibrium is stable. This is the case when the body is loaded at the lowest parts, and it is clear that the action of each will be to bring g to the lowest position it can take.

When m is below g, then, the fluid pressing upwards, and the weight acting downwards, will each tend to move the body farther and farther from its first position, so that the equilibrium is unstable.

When m coincides with g, then the forces being equal and opposite, and acting at the same point, the body will rest in that position, and the equilibrium will be neutral Flotation, or indifferent.

90. Cor.—Hence the necessity of having the heavier Corollary goods of a ship's cargo stowed away at the bottom of the vessel, and of having the ship properly ballasted, or the keel well laden. For the masts and rigging may tend to raise the centre of gravity of the ship above the metacentre, in which case the vessel would be thrown on her beam ends. Similarly, also, it is much more safe to sit in a small boat than to stand upright, for the centre of gravity of the craft is low in the former, but high in the latter case.

SECTION II.—SPECIFIC GRAVITIES.

91. Definition.—By the specific gravity of a substance is meant the ratio subsisting between the weights of equal gravities, volumes of that substance and some other known substance taken as a standard.

92. It has been agreed among philosophers to take as the standard of comparison the purest distilled water, at a temperature of 60° Fahr., and when the barometer is at 30 inches; the values or quotients of the ratios expressing the specific gravities of different substances, being all set down in a table of specific gravities. Thus the weight of a cubic foot of pure platina is 19,500 ounces, but the weight of a cubic foot of standard water is 1000 ounces; therefore the ratio between the former and latter numbers, or $\frac{19500}{1000} = 19.5$ is the specific gravity of pure platina—that is to say, since all matter is equally heavy, that there are 19.5 times more matter in a cubic foot of platina than there is in a cubic foot of pure distilled water. If, then, unity or 1 be taken as the specific gravity of water, 19.5 will be that of pure platina, or we have only in the table to count off from right to left three places, so as to obtain the specific gravity of the several substances there set down.

93. With respect to the specific gravities of the gases, on the other hand, they may be reckoned in terms of water or atmospheric air; in either case we shall have the same results. Biot and Arago have deduced from very accurate experiments, that when the temperature was at 59° Fahr., air was 800 times lighter than water, and when at the freezing point, it was 770 times lighter. The weight of a cubic foot of air, then, being taken as 1 ounce, 1000 such volumes will weigh 1000 ounces. But it is found that 1000 cubic feet of hydriodic acid weighs 4300 ounces. Hence the specific gravity of hydriodic acid is the ratio between the latter and the former numbers, or $\frac{4300}{1000} = 4.3$.

94. Prop. V.—The densities of different substances are as their specific gravities.

Let A and B be the bodies, having volumes V and V', and densities $\rho$ and $\rho'$ respectively; then, A weighs $V\rho g$, while B weighs $V'\rho'g$. If the substances be compared with the standard fluid, the density of which, like the specific gravity, is unity, the weights of volumes V and V' of it will be $Vg$ and $V'g$; therefore,

$$\text{Sp.gr.of } A = \frac{V\rho g}{V'g}, \quad \text{and sp.gr.of } B = \frac{V'\rho'g}{V'g};$$

or, Sp.gr.of $A : \text{sp.gr.of } B = \frac{V\rho g}{V'g} : \frac{V'\rho'g}{V'g} = \rho : \rho'$.

95. Cor.—The specific gravity of different substances Corollary will be as the weights of equal bulks of the substances.

96. Prop. VI.—If a body be immersed in a fluid, the weight lost is to the weight of the body, as the specific gravity of the fluid is to the specific gravity of the body. Suppose that \( W \) is the weight of the body, and \( W' \) that lost by immersion; then \( W - W' \) is the weight by which it sinks, since \( W' \) is the weight of the fluid displaced. Let \( V \) be the volume of the body, \( \rho \) and \( \rho' \) the density of the body and fluid respectively; then

\[ W = V\rho g, \quad W' = V\rho' g. \]

\[ \therefore \frac{W}{W'} = \frac{\rho}{\rho'}. \]

But (by Prop. V.)

\[ \text{Specific gr. of fluid} : \text{specific gr. of body}. \]

97. Prop. VII.—To find the specific gravity of a substance.

Let \( W \) be the weight of the body in air, \( W' \) its weight in water; then weight lost \( = W - W' \); weight of fluid displaced, and the volume displaced \( = \) that of the body. Hence \( W - W' \) is the weight of a volume of fluid equal to the volume of the body. Therefore, by def. 92,

\[ \text{Specific gravity of substance} = \frac{W}{W - W'}. \]

Let the body be lighter than water; then, in order to make it sink, let a different body be attached to it, the weight of which is \( W_1 \) in air, and \( W_2 \) in water. Let also \( W' \) be the weight of both bodies in the water. Then, \( W \) being weight of body,

Weight of water equal in volume to both bodies \( = W + W_1 - W' \)

Weight of water equal in volume to attached body \( = W_1 - W_2 \)

\[ \therefore \text{Weight of water equal in volume to the body the specific gravity of which is required} = W_1 + W_2 - W'. \]

Specific gravity of body \( = \frac{W}{W_1 + W_2 - W'} \).

If the body be soluble, as sugar, &c., in water, it must be inclosed in wax or some other envelope, and a process similar to the last case will determine its specific gravity.

98. Prop. VIII.—To find the specific gravity of a compound body. All bodies of the same material have always the same specific gravity; but a compound body has a specific gravity different from that of either of its components.

Let \( V \) and \( V' \) be the volumes of the components, \( \rho \) and \( \rho' \) their densities, \( \rho'' \) that of the compound; then

\[ (V + V') \rho'' = V\rho + V'\rho'; \quad \text{or}, \quad \frac{(V + V') \rho''}{\rho} = \frac{V\rho}{\rho} + \frac{V'\rho'}{\rho}, \]

where \( \rho = \text{unity or } 1; \quad \text{or}, \quad (V + V') S'' = VS + VS', \quad \text{since specific gravities are as their densities (94)}; \]

\[ \therefore \text{Specific gravity of compound} = \frac{VS + VS'}{V + V'}. \]

99. It is said that Hiero, king of Syracuse, gave a goldsmith a quantity of pure gold to be made into a crown. On the crown being brought to him and examined, Hiero suspected that the goldsmith had adulterated the pure metal; he therefore inquired at Archimedes if his suspicions could be verified or disproved without injuring the crown. Archimedes, shortly after, showed Hiero that the crown was alloyed with a base metal. The adulteration must have been detected by the principles already explained.

100. Prop. IX.—To find the specific gravity of a fluid.

Let a vessel be filled with the fluid, and, being weighed in air, let its weight be \( W \), and let \( W' \) be the weight of the vessel when empty; let also the same vessel when filled with pure distilled water, and weighed in air, be \( W'' \); if care be taken, the volumes of the fluid and of the water will be the same. Now the volume of the fluid weighs \( W - W' \); while an equal volume of water weighs \( W'' - W' \);

\[ \therefore \text{Specific gravity of fluid} = \frac{W - W'}{W'' - W'}. \]

101. Cor.—After the same manner the specific gravity of atmospheric air, or any of the gases, or even fine powders, may be obtained.

102. Prop. X.—The specific gravities of two fluids may be compared by weighing the same solid in each.

Let the body weigh \( W \) in air, \( W_1 \) when immersed in the first fluid, \( W_2 \) in the second; then,

Weight of first fluid displaced \( = W - W_1 \), and

Weight of second fluid displaced \( = W - W_2 \).

But each of these expressions is the weight of equal volumes of different substances; therefore (Cor. 95) if \( S_1 \) and \( S_2 \) be the specific gravities of the first and second fluid;

Sp. gr. of first fluid : sp. gr. of second fluid \( = W - W_1 : W - W_2 \);

or, \( S_1 : S_2 = W - W_1 : W - W_2 \); or, \( S_1 = \frac{W - W_1}{W - W_2} \).

103. Prop. XI.—When several fluids are thrown together into the same vessel, they will either become a compound fluid, or remain unmixed, and superposed above each other, their surfaces being horizontal.

Let the fluids have volumes \( V_1, V_2, V_3, \ldots, V_n \), respectively, and corresponding specific gravities \( S_1, S_2, S_3, \ldots, S_n \); then, when the fluids mix together, the specific gravity of the new fluid will be found by art. 98 to be

\[ S = \frac{V_1 S_1 + V_2 S_2 + V_3 S_3 + \ldots + V_n S_n}{V_1 + V_2 + V_3 + \ldots + V_n}. \]

104. If the fluids do not mix, then they will range themselves according to their specific gravities, and equilibrium will be attained when the common surface of any two of them is horizontal. For suppose that we have a vessel \( M \) (fig. 13), containing say two fluids, the heavier \( AB \) lowest, and its surface assuming the outline \( AB \), and a lighter fluid above it. Take \( CAE \) and \( DBF \), vertical lines, \( C \) and \( D \) being points in the horizontal surface of the lighter fluid, \( A \) and \( B \) points on the surface of the heavier. \( E \) and \( F \) horizontal points in the depth of the same fluid. Now, it is evident, that at points \( C \) and \( D \), on the horizontal surface of the upper fluid, the pressures are equal; and if the same fluid had filled the vessel, those at \( E \) and \( F \) would also have been equal, being caused by the weight of a column extending from \( F \) or \( E \) to the surface. But the pressures at \( A \) and \( B \) of the heavier fluid are unequal, the former being pressed by a lighter column than the latter; and although the cylindrical columns \( DF, CE \) are of the same dimensions throughout, yet in the latter there is a greater quantity of the heavier fluid than in the former; the difference of weights of the columns cannot be compensated by the less heavy fluid, and consequently the pressures at the points \( E \) and \( F \) are unequal. Hence equilibrium cannot obtain; for before an equilibrium can subsist, it is necessary and sufficient that the points in the same horizontal plane should be pressed equally; this can only obtain when the surface \( AB \) is truly horizontal. It is not necessary that the differently dense and immiscible fluids should range themselves in the vessel according to their weights. It is quite possible for a heavy fluid to be uniformly distributed over the surface of a lighter fluid, but in such a position, the equilibrium would be unstable, and the slightest displacement of the vessel from one side to the other would cause the heavy fluid to descend and assume the lowest position in the vessel.

If, then, several fluids of different densities, and which do not mix, be thrown into a vessel, and the vessel shaken for any length of time, and then allowed to stand, the fluid Of specific particles of the same kind will gradually settle in the order of their specific gravities, and with plane horizontal surfaces.

**SECTION III.—ON THE HYDROMETER.**

105. In order to determine with expedition the strength of spirituous liquors, which are inversely proportional to their specific gravities, an instrument more simple, though less accurate, than the hydrostatic balance, has been generally employed. This instrument is called a hydrometer, sometimes an areometer and gravimeter, and very erroneously a hygrometer by some foreign authors. It seems to have been invented by Hypatia, the daughter of Theon Alexandrinus, who flourished about the end of the fourth century; though there is some foundation for the opinion that the invention is due to Archimedes.

106. The Common Hydrometer was invented by Fahrenheit, and is the simplest of this class of instruments. It consists of two hollow spheres, C and D (fig. 14); BEE is a delicate cylindrical stem, nicely graduated; and D is loaded with mercury or lead, that the centre of gravity of the instrument may be below that of the fluid displaced.

Let \( V = \) volume of this Common Hydrometer and \( k \) the section of the graduated stem. Let the instrument be placed in a fluid A of a density \( \rho \), and let it sink to a depth E, while in another fluid, B, of a density \( \rho' \), it sinks to F; let also the depths of E and F from the zero point, or top of the stem, be \( x \) and \( y \) divisions respectively. Then, volume of fluid displaced of A \( = (V - xk) \), and volume of fluid displaced of B \( = (V - yk) \); the weights of which are respectively \( (V - xk)\rho \), and \( (V - yk)\rho' \).

But as the weight of the fluid displaced is equal to the weight of the body immersed, we have \((V - xk)\rho = (V - yk)\rho' \); or, \((V - xk) : (V - yk) = \rho : \rho' \); \( \rho \) = specific gravity of A; \( \rho' \) = specific gravity of B (94).

\[ \frac{V - yk}{V - xk} = \text{specific gravity of } A = \text{specific gravity of } B \cdot \frac{V - yk}{V - xk} \]

\( V - yk = V - xk \), since B is regarded as the standard fluid of which the specific gravity is unity or 1.

107. Jones's Hydrometer (fig. 16) is a simple and accurate instrument, requiring only three weights to discover the strengths of spirituous liquors from alcohol to water. It is adjusted to the temperature of 60° Fahr., and has an attached thermometer, so as to make due allowance for a variation of the temperature.

In fig. 16, the whole instrument is represented with the thermometer attached to it. Its length, AB, is about 9½ inches; the ball C is made of hard brass, and nearly oval, having its conjugate diameter about 1½ inches. The stem AD is a parallel-piped, on the four sides of which the different strengths of spirits are engraved: the three sides which do not appear in fig. 16 are represented in fig. 15, with the three weights numbered 1, 2, 3, corresponding with the sides similarly marked at the top. If the instrument, when placed in the spirits, sinks to the divisions on the stem without a weight, their strength will be shown on the side AD marked at the top, and any degree of strength from 74 gallons in the 100 above proof, will thus be indicated.

If the hydrometer does not sink to the divisions without a weight, it must be loaded with any of the weights 1, 2, 3, till the ball C is completely immersed. If the weight No. 1 is the necessary, the side marked I will show the strength of spirits, from 46 to 13 gallons in the 100 above proof. If the weight No. 2 is employed, the corresponding side will indicate the remainder of over proof to proof, marked P in the instrument, and likewise every gallon in 100 under proof, down to 29. When the weight No. 3 is used, the side similarly marked will show any strength from 30 gallons in the 100 under proof, down to water, which is marked W in the scale. The small figures, as 4 at 66, 3 at 61, 2½ at 48 (fig. 16) indicate the diminution of bulk which takes place when water is mixed with spirits of wine in order to reduce it to proof; thus, if the spirit be 61 gallons in the 100 over proof, and if 61 gallons of water are added in order to render it proof, the magnitude of the mixture will be 3½ gallons less than the sum of the magnitudes of the ingredients; that is, instead of being 161 it will be only 157½ gallons. The thermometer F connected with the hydrometer, has four columns engraved upon it, two on one side as seen in the figure, and two on the other side. When any of the scales upon the hydrometer, marked 0, 1, 2, 3, are employed, the column of the thermometer similarly marked must be used, and the number at which the mercury stands carefully observed. The divisions commence at the middle of each column which is marked 0, and is equivalent to a temperature of 60° of Fahrenheit; then, whatever number of divisions the mercury stands above the zero of the scale, the same number of gallons in the 100 must the spirit be reckoned weaker than the hydrometer indicates, and whatever number of divisions the mercury stands below the zero, so many gallons in the 100 must the spirit be reckoned stronger.

108. Nicholson's Hydrometer (fig. 17) serves to determine the specific gravity of both solids and fluids. A hollow light body, A, is pierced by a graduated delicate stem D, carrying a dish, C, for weights, and a cup, B, loaded so as to insure a stable equilibrium in the instrument. The instrument is always sunk to the same depth by weights placed in the upper dish.

1. To find the specific gravity of a body with this instrument, let \( W \) be the weight of the hydrometer, \( W' \) weight of fluid displaced when the stem is at D, \( w \) that weight placed in the dish so as to sink it to this depth. Let also \( X \) be the weight of the body, the specific gravity of which is required, and \( Y \) the weight of a corresponding volume of fluid equal to the bulk of the body. Then,

\[ \text{Specific gravity of solid} = \frac{X}{Y} \]

Let the body be put first in the upper cup, and let \( w' \) be the weight required to sink the instrument to D; then, when the body is not put on C, we have \( W' = W + w' \); but the body being on C, we have also \( W' = W + X + w' \); \( X = w' - w \).

Now, let the body be placed in the lower cup B, and let a weight \( w'' \) sink the instrument to D; then again, \( W' + Y = W + X + w'' \), from which if we subtract \( W' = W + X + w'' \), gives us \( Y = w' - w'' \);

\[ \text{Specific gravity of body} = \frac{X}{Y} = \frac{w' - w}{w'' - w} \]

2. In order to compare the specific gravity of two fluids with this instrument, let again \( W \) be its weight, and \( w \) that re- Weight of the displaced fluid of \( A = W + w \) and Weight of the displaced fluid of \( B = W + w' \).

But these two weights being the same, and the specific gravities of two fluids being as the ratio of equal bulks of them; \( \therefore \text{Sp.gr. of } A : \text{sp.gr. of } B = W + w : W + w' \). Let \( B \) be the standard fluid, and specific gravity of \( A = \frac{W + w}{W + w'} \).

109. The Barometrical Areometer is an instrument which is more useful for the purposes of illustration than of measurement. If two immiscible liquids are poured into a two-branched tube \( ACB \), the one into the branch \( BC \), and the other into the branch \( AC \), till they balance each other, their specific gravities will be to one another inversely as the heights of each column.

Thus, if we pour in mercury at \( A \), and water at \( B \), so that when the surface of the mercury is at \( D \), that of the water is at \( E \), we shall find that if the column of mercury \( DF \) is two inches, that of the water \( EG \) will be 27 inches, and their specific gravities will be as 27 to 2, or as 13½ to 1. If we pour in at \( B \) linseed oil in place of water, the height \( EG \) will be 29 inches, and the specific gravity of the oil \( 0.931 \); because \( 27 \text{ ft.} : 2 \text{ in.} = 13\frac{1}{2} : 0.931 \). By thus using mercury as the balancing column, the specific gravities of all fluids that do not mix with it, or act upon it, may be readily ascertained. The results thus obtained are not affected by the admission of the air at the open ends \( A \) and \( B \), because the same weight of air presses upon the two balancing columns. But if we pour in mercury at \( A \) till the bent tube \( ACB \) contains above thirty inches of it, and close up the end \( A \), and remove the air from above the mercury in \( AC \), the column of mercury, being no longer pressed down by the air in \( AC \), will be pressed up to near the top of the tube \( A \) by the pressure of the column of air in \( BC \), and the instrument becomes a barometer, a column of air balancing a column of mercury. In this case, the tube \( BI \) becomes unnecessary, and the mercury may be inclosed in a glass ball at \( I \), with an opening to admit the air.

110. Say's Stereometer (fig. 19) is for the purpose of determining the volumes, and from these the specific gravity of liquid bodies, soft bodies, porous bodies, and powders, as also solids. The instrument consists of a glass tube of uniform bore, ending in a cup \( PE \), the mouth being ground truly plane, and capable of being rendered air-tight by the plate of glass \( E \). The cup \( PE \) contains the substance \( B \), the volume of which is wanted. Take off the covering plate \( E \), and immerse the tube \( PC \) in a vessel \( D \) of mercury till the mercury reaches \( P \). Now cover the cup with the plate \( E \), and raise \( PC \) till the mercury is at \( M \), and its level in \( D \) at \( C \); the height of the mercurial column in \( PC \) is now \( CM \).

As there is still so much air in \( PE \), let its volume be \( u \) before \( B \) was put in, and \( v \) that of the solid in the cup, \( k \) the horizontal section of \( PC \), \( h \) the altitude of the mercurial column in \( PC \), and \( \rho \) the density of mercury. When the column reached \( P \), the volume of air \( = u - v \), and its pressure \( = \rho gh \); when again it was at \( M \), and in \( D \) at \( C \), the air in \( EP \) occupied a space \( = u - v + k \cdot PM \), and its pressure \( = \rho h \cdot CM \). Since, of specific gravity, the pressure of the air varies inversely as the space it occupies, we have \( u - v + k \cdot PM : u - v = h : h - CM \);

\[ e = \frac{h - CM}{MC} \cdot k \cdot PM. \]

So also \( u \) may be found, the cup being empty; \( k \) will be found by weighing the mercury filling a certain portion of the tube. Thus, since a cubic inch of mercury weighs at \( 60^\circ \text{ Fahr.} \) 34291 grains nearly; therefore, if the mercurial column in \( PC \) be \( a \) inches, and its weight \( w \) grams, \( w = 34291 \times (a \cdot \text{volume of mercury in } PC) = (34291 \times k \cdot a) \), where \( k \) is an area or square inches.

If then the solid \( B \) be a weight \( = w \), its specific gravity \( = \frac{W}{v} \).

The specific gravity of powders and soluble substances may be found in the same manner.

111. The Stakometer, or Drop-measurer, is shown in fig. 20, where \( ABC \) is a glass vessel four or five inches long, having a hollow bulb, \( B \), about half an inch in diameter. The instrument is filled by suction, and the fluid is discharged at \( C \) till it stands nearly at the point \( m \), the zero of the scale. The fluid is then allowed to discharge itself at \( C \) by drops, and the number of them is counted till the surface of the fluid descends to another fixed point \( n \). The experiment is then carefully repeated at different temperatures, till the number of drops of distilled water occupied by the cavity between \( m \) and \( n \) is accurately determined for various temperatures. The same experiment is made with alcohol. Thus, if \( N \) is the number of drops of distilled water whose specific gravity is \( S \), and \( n \) the number of drops of alcohol whose specific gravity is \( s \), and \( d \) the number of drops of any other mixture of alcohol and water contained in the same cavity \( mn \), we shall have

\[ n - N : S - s = d - N : \frac{(d - N)(S - s)}{n - N}; \]

and therefore

\[ S - \frac{(d - N)(S - s)}{n - N} \]

will be the specific gravity of the mixture required.

With a small instrument, the number of drops of water between \( m \) and \( n \) was 724, whereas the number of drops of ordinary proof spirits was 2117 at \( 60^\circ \text{ Fahr.} \). Now, as the specific gravity of the spirits was .920, and that of water 1.000, we have a scale of 1395 drops for measuring all specific gravities between .920 and 1.000, an unit in the fourth place of decimals corresponding to a variation of about two drops. From this experiment it follows that the bulk of a drop of water will be about 2.93 times as large as the bulk of a drop of the spirits.

112. Sikes' Hydrometer (fig. 21) is used in the collection of the revenue of the United Kingdom; it determines the relative quantity of alcohol and water which wine and spirituous liquors contain. If in the spirit to be tested there be more water than alcohol, the hydrometer will show that it is below proof; if the contrary, above proof. The only difference between Sikes' and the common hydrometer is that the stem of this is thin and flat; it has besides eight small weights which may be placed on the lower stem, \( D \), so as to increase the weight of the instrument, since the specific gravity of light fluids would prevent it from sinking without the addition of one or more of these weights to the lowest division on the graduated stem in a heavy fluid. Let \( V \) = volume of the instrument, \( W \) its weight, \( K \) the area of a section of CE. When placed in the fluid A let X be the weight at C, and P the surface of the fluid; when it floats in B, let Y be the weight at C, and Q the surface of the fluid, and let R, S be the respective volumes of X and Y; then,

\[ \begin{align*} \text{Weight of fluid } A & = W + X, \\ \text{Weight of fluid } B & = W + Y, \\ \text{Volume of fluid } A & = V + R - K \cdot CP, \\ \text{Volume of fluid } B & = V + S - K \cdot CQ. \end{align*} \]

\[ \begin{align*} W + X & = \text{specific gravity of } A \times (V + R - K \cdot CP), \\ W + Y & = \text{specific gravity of } B \times (V + S - K \cdot CQ); \end{align*} \]

\[ \begin{align*} (W + X)(V + S - K \cdot CQ) & = (W + Y)(V + R - K \cdot CP), \end{align*} \]

the standard fluid being B.

### SECTION IV.—ON TABLES OF SPECIFIC GRAVITIES.

113. As the knowledge of the specific gravities of bodies is of great use in all the branches of mechanical philosophy, we have given the following table, computed by Mr Tod, civil engineer, and published in the second edition of his excellent work, entitled *Series of Tables*, and which ought to be in the hands of every engineer, architect, and country gentleman:

#### The Specific Gravity of Bodies calculated to Avoirdupois Weight.

| Name of Bodies | Weight of a Cubic foot in ozs. and lbs. | Weight of a Cubic inch in ozs. | Number of Cubic inches in a lb. | |----------------|----------------------------------------|-------------------------------|-------------------------------| | METALS | | | | | Antimony, cast.| 6702 | 418-8750 | 3-6748 | | Zinc, cast. | 7190 | 449-3750 | 3-6431 | | Iron, cast. | 7267 | 459-8375 | 3-4707 | | Tin, cast. | 7921 | 455-8875 | 3-4293 | | Tin, hardened. | 7299 | 456-1875 | 3-4239 | | Pewter. | 7471 | 460-9375 | 3-4324 | | Iron, bar. | 7788 | 486-7500 | 3-5699 | | Cobalt, cast. | 7811 | 488-1875 | 3-5292 | | Steel, hard. | 7816 | 488-5000 | 3-5231 | | Steel, soft. | 7833 | 489-5625 | 3-5229 | | Iron, meteoric, hammered. | 7965 | 497-8125 | 3-4792 | | Nickel, cast. | 8279 | 517-4375 | 3-3945 | | Brass, cast. | 8395 | 524-6875 | 3-2833 | | Brass, pure. | 8344 | 534-0000 | 3-1944 | | Nickel, hammered. | 8666 | 541-6250 | 3-0150 | | Gun metal. | 8784 | 549-0000 | 3-0833 | | Copper, cast. | 8788 | 549-2500 | 3-0856 | | Copper, wire. | 8878 | 554-8750 | 3-0377 | | Copper, coin. | 8915 | 557-0175 | 3-0151 | | Bismuth, cast. | 9822 | 613-8750 | 2-6840 | | Silver, hammered. | 10610 | 656-8750 | 2-6021 | | Silver, coin. | 10634 | 658-3750 | 2-6240 | | Silver, pure. | 11000 | 687-5000 | 2-5134 | | Rhodium. | 11000 | 687-5000 | 2-5134 | | Lead, cast. | 11352 | 709-5000 | 2-3694 | | Palladium. | 11800 | 737-5000 | 2-0827 | | Mercury (quicksilver), common. | 13568 | 848-0000 | 2-0377 | | Mercury (do.), pure. | 14000 | 875-0000 | 2-0108 | | Gold, trinket. | 15709 | 981-8125 | 1-9008 | | Gold, coin. | 17647 | 1102-9375 | 1-0124 | | Gold, pure, cast. | 19258 | 1303-6250 | 1-1446 | | Gold, hammered. | 19316 | 1216-6875 | 1-2042 | | Platinum, pure. | 19350 | 1312-7500 | 1-1417 | | Platinum, hammered. | 20141 | 1315-0625 | 1-2175 | | Platinum, wire. | 22069 | 1379-8125 | 1-2714 | | Iridium, hammered. | 23000 | 1437-5000 | 1-3101 | | EARTH, STONES, &c. | | | | | Amber. | 1678 | 67-3750 | 0-6284 | | Coal. | 1250 | 78-7500 | 0-72337 | | Sand. | 1500 | 93-7500 | 0-86303 | | Brick. | 2000 | 125-0000 | 1-05740 | | Sulphur, native. | 2033 | 127-0625 | 1-17650 | | Opal. | 2114 | 132-1250 | 1-22337 | | Clay. | 2160 | 135-0000 | 1-25000 | | Gypsum. | 2241 | 140-3125 | 1-35474 | | Porcelain, Limoges. | 2341 | 146-3125 | 1-35474 | | Porcelain, China. | 2395 | 147-2500 | 1-38020 | | Stone, paving. | 2416 | 151-4000 | 1-39614 | | Gunpowder, loose. | 836 | 52-2500 | 0-48379 | | Living men. | 891 | 55-6875 | 0-51562 | | Wax. | 897 | 56-0625 | 0-51900 | | Ice. | 930 | 58-1250 | 0-53810 | | Gunpowder, close. | 937 | 58-5625 | 0-54224 | | Tallow. | 942 | 58-6750 | 0-54512 | | Butter. | 942 | 58-8750 | 0-54513 | | Bees-wax. | 956 | 59-7500 | 0-55324 | | Sodium. | 972 | 60-7500 | 0-56250 |

RESINS, GUMS, &c.

| Gunpowder, heap. | 836 | 52-2500 | 0-48379 | | Living men. | 891 | 55-6875 | 0-51562 | | Wax. | 897 | 56-0625 | 0-51900 | | Ice. | 930 | 58-1250 | 0-53810 | | Gunpowder, close. | 937 | 58-5625 | 0-54224 | | Tallow. | 942 | 58-6750 | 0-54512 | | Butter. | 942 | 58-8750 | 0-54513 | | Bees-wax. | 956 | 59-7500 | 0-55324 | | Sodium. | 972 | 60-7500 | 0-56250 | ### Hydrodynamics

#### The Specific Gravity of Bodies calculated by Avoirdupois Weight.—Continued.

| Name of Bodies | Weight of a Cubic foot in ozs. and lbs. | Weight of a Cubic inch in ozs. | Number of Cubic inches in a lb. | |----------------|----------------------------------------|-------------------------------|--------------------------------| | **Resins, Gums, &c.** | | | | | Camphor | 989 | 61-8125 | 0-56655 | 27-9555 | | Rosin | 1100 | 68-7000 | 0-63557 | 25-0909 | | Pitch | 1150 | 71-8750 | 0-66550 | 24-9417 | | Opium | 1337 | 83-5625 | 0-77372 | 20-6791 | | Gum Arabic | 1452 | 90-7500 | 0-84027 | 19-0413 | | Honey | 1456 | 91-0000 | 0-84259 | 18-9890 | | Bone of an Ox | 1659 | 103-6875 | 0-96006 | 16-6654 | | Bone, dry | 1660 | 103-7500 | 0-96084 | 16-6554 | | Flaxseed | 1714 | 107-1250 | 0-99184 | 16-1307 | | Alum | 1745 | 109-0625 | 1-00833 | 15-8411 | | Gunpowder, solid | 1900 | 118-7500 | 1-09933 | 14-5515 | | Nitre (saltpetre) | 1917 | 119-8125 | 1-10937 | 14-4422 | | Ivory | | | | |

| **Woods.** | | | | | Cork | 240 | 15-0000 | 0-13838 | 115-2000 | | Poplar | 383 | 23-9375 | 0-22104 | 71-7960 | | Larch | 544 | 34-0000 | 0-31481 | 50-8235 | | Fir, North of England | 556 | 34-7500 | 0-32175 | 49-7265 | | Mahogany, Honduras | 560 | 35-0000 | 0-32407 | 49-3714 | | Cedar, American | 561 | 35-0625 | 0-32465 | 49-2833 | | Poon | 579 | 36-1875 | 0-33506 | 47-7512 | | Willow | 585 | 36-5625 | 0-33854 | 47-2013 | | Cedar | 598 | 37-2500 | 0-34490 | 46-3892 | | Cypress | 598 | 37-2500 | 0-34494 | 46-2341 | | Birch | 600 | 37-5000 | 0-34664 | 46-0000 | | Pitch Pine | 600 | 41-2500 | 0-38194 | 41-8099 | | Pear-tree | 661 | 41-3125 | 0-38292 | 41-8275 | | Walnut | 681 | 42-5625 | 0-39457 | 40-5991 | | Mar Forest Fir | 694 | 43-3750 | 0-40162 | 39-3386 | | Elder-tree | 695 | 43-4375 | 0-40219 | 39-7812 | | Orange-tree | 705 | 44-0625 | 0-40798 | 39-2170 | | Cherry-tree | 715 | 44-6875 | 0-41377 | 38-6885 | | Teak | 745 | 46-5625 | 0-43113 | 37-1114 | | Riga Fir | 750 | 47-0000 | 0-43402 | 36-8462 | | Maple | 755 | 47-1875 | 0-43677 | 36-7438 | | Oak, Danish | 768 | 47-5000 | 0-43861 | 36-3789 | | Yew, Dutch | 788 | 49-2500 | 0-45590 | 35-0862 | | Apple-tree | 793 | 49-5625 | 0-45891 | 34-8556 | | Yew, Spanish | 807 | 50-4375 | 0-46701 | 34-2602 | | Ash | 845 | 52-8125 | 0-48900 | 32-7195 | | Beech | 852 | 53-2500 | 0-49305 | 32-4597 | | Oak, Canadian | 872 | 54-5000 | 0-50694 | 31-7064 | | Logwood | 913 | 57-0625 | 0-53125 | 30-2825 | | Oak, English | 970 | 60-6250 | 0-56134 | 28-5030 |

| **Gases.** | | | | | Hydrogen | 0-069 | 0-043125 | -0-000399 | 400-995-6 | | Ammonia | 0-559 | 0-368750 | -0-003414 | 46861-0 | | Nitrogen | 0-972 | 0-607500 | -0-005625 | 28444-4 | | Olefiant | 0-982 | 0-613750 | -0-005682 | 28154-7 | | Atmospheric air | 1-000 | 0-625000 | -0-005787 | 27648-0 | | Nitrous | 1-042 | 0-651250 | -0-006030 | 26533-5 | | Oxygen | 1-111 | 0-687500 | -0-006283 | 26088-6 | | Mariatic acid | 1-200 | 0-700000 | -0-006467 | 25600-0 | | Carbonic acid | 1-524 | 0-952500 | -0-008819 | 18141-7 | | Cyanogen | 1-805 | 1-128125 | -0-010445 | 15317-4 | | Sulphurous acid | 2-222 | 1-388750 | -0-012858 | 12442-8 | | Chlorine | 2-444 | 1-527500 | -0-014143 | 11312-6 | | Fluosilicic acid | 3-011 | 2-258750 | -0-020896 | 7656-6 | | Hydrochloric acid | 4-300 | 2-687500 | -0-024884 | 6129-7 |

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*Vol. XII.* CHAPTER III.—ON CAPILLARY ATTRACTION AND THE COHESION OF FLUIDS.

114. We have already seen, when discussing the equilibrium of fluids, that when water or any other fluid is poured into a vessel, or any number of communicating vessels, its surface will be horizontal, or it will rise to the same height in each vessel, whatever be its form or position. This proposition, however, only holds true when the diameter of these vessels or tubes exceeds the fifteenth of an inch; for if a system of communicating vessels be composed of tubes of various diameters, the fluid will rise to a level surface in all the tubes which exceed one fifteenth of an inch in diameter; but in the tubes of a smaller bore, it will rise above that level to altitudes inversely proportional to the diameters of the tubes. The power by which the fluid is raised above its natural level is called capillary attraction, and the glass tubes which are employed to exhibit this phenomena are named capillary tubes. These appellations derive their origin from the Latin word capillus, signifying a hair, because the bores of these tubes have the fineness of a hair.

115. When we bring a piece of clean glass in contact with water or any other fluid, except mercury and fused metals, and withdraw it gently from its surface, a portion of the fluid will not only adhere to the glass, but a small force is necessary to detach this glass from the fluid mass, which resists any separation of its parts. Hence it is obvious that there is an attraction of cohesion between glass and water, and that the constituent particles of water have also an attraction for each other. The suspension of a drop of water from the lower side of a plate of glass is a more palpable illustration of the first of these truths; and the following experiment will completely verify the second. Place two large drops of water on a smooth metallic surface, their distance being about the tenth of an inch. With the point of a pin unite these drops by two parallel canals, and the drops will instantly rush to each other through these canals, and fill the dry space that intervenes. This experiment shows that in capillary attraction there enters an attractive as well as a cohesive force.

116. Upon these principles many attempts have been made to account for the elevation of water in capillary tubes; but most of the explanations which have hitherto been offered, are founded upon hypothesis, and are very far from being satisfactory. Without presuming to substitute a better explanation in the room of those which have been already given, and so frequently repeated, we shall endeavour to illustrate that explanation of the phenomena of capillary attraction which seems liable to the fewest objections. For this purpose let a drop of water be laid upon a horizontal glass plate. Every particle of the glass immediately below that drop exerts an attractive force upon the particles of water. This force will produce the same effect upon the drop as a pressure in the opposite direction; the pressure of a column of air, for instance, on the upper surface of the drop. The effect of the attractive force, therefore, tending to press the drop to the glass will be an enlargement of its size, and the water will occupy a larger space; this increase of its dimensions will take place when the surface AB is held downwards; and that it does not arise from atmospheric pressure may be shown by performing the experiment in vacuo. Now, let AB (fig. 22) be a section of a plate of glass, which is held vertically, part of the water will descend by its gravity, and form a drop B, while a small film of the fluid will be supported at m by the attraction of the glass. Bring a similar plate of glass CD, into a position parallel to AB, and make them approach nearer and nearer each other. When the drops B and D come in contact, they will rush together from their mutual attraction, and will fill the space op. The gravity of the drops B and D being thus diminished, the films of water at m and n, which were prevented from rising by their gravity, will move upwards. As the plates of glass continue to approximate, the space between them will fill with water, and the films at m and n being no longer prevented from yielding to the action of the glass immediately below them (by the gravity of the water at op, which is diminished by the mutual action of the fluid particles), will rise higher in proportion to the approach of the plates. Hence it may be easily understood how the water rises in capillary tubes, and how its altitude is inversely as their internal diameters. For let A, a be the altitudes of the fluid in two tubes of different diameters, D, d; and let C, c be the two cylinders of fluid which are raised by virtue of the attraction of the glass. Now, as the force which raises the fluid must be as the number of attracting particles, that is, as the surface of the tube in contact with the water, that is, as the diameter of the tubes; and as this same force must be proportional to its effects on the cylinder of water raised, we shall have $D : d = C : c$; But (Geometry, sect. viii., theor. xi.; sect. ix., theor. ii.) $C : c = D'A : d'a$, therefore $D'A : d'a = D : d$; hence $DA \cdot d = d'a \cdot D$, and $DA = \frac{d'aD}{D}$; or, $DA = da$, that is, $D : d = a : A$; or the altitudes of the water are inversely as the diameters of the tubes. Since $DA = da$, the product of the diameter by the altitude of the water will always be a constant quantity. In a tube whose diameter is 001, or 100th of an inch, the water has been found to reach the altitude of 53 inches; hence the constant quantity $53 \times 01 = 0053$ may fitly represent the attraction of glass for water. According to the experiments of Muschenbroek, the constant quantity is 0059; according to Weitbrecht, 00428; according to Monge, 0042; and according to Atwood, 00530. When a glass tube was immersed in melted lead, Gellert found the depression multiplied by the bore to be 0054.

117. Having thus attempted to explain the causes of capillary action, we shall now proceed to consider some of its more interesting phenomena. In fig. 23, MN is a vessel of water in which tubes of various forms are immersed. The water will rise in the tubes A, B, C, to different altitudes, m, n, o, inversely proportional to their diameters. If the tube B is broken at a, the water will not rise to the very top of it at a, but will stand at b, a little below the Capillary top, whatever be the length of the tube, or the diameter of attraction, its bore. If the tube be taken from the fluid and laid in a horizontal position, the water will recede from the end that was immersed.

118. If a tube D, composed of two cylindrical tubes of different bores, be immersed in water with the widest part downwards, the water will rise to the altitude p; and if another tube E of the same size and form be plunged in the fluid with the smaller end downwards, the water will rise to the same height q as it did in the tube D. This experiment seems to be a complete refutation of the opinion of Dr Jurin, that the water is raised by the action of the annulus of glass above the fluid column; for since the annular surface is the same at q as at p, the same quantity of fluid ought to be supported in both tubes, whereas the tube E evidently raises much less water than D. But if we admit the supposition in art. 116, that the fluid is supported by the whole surface of glass in contact with the water, the phenomenon receives a complete explanation; for since the surface of glass in contact with the fluid in the tube E is much less than the surface in contact with it in the tube D, the quantity of fluid sustained in the former ought to be much less than the quantity supported in the latter.

119. When a vessel F (fig. 23), is plunged in water, and the lower part, ture, filled by suction till the fluid enter the part F4, the water will rise to the same height as it does in the capillary tube G, whose bore is equal to the bore of the part F4. In this experiment the portions of water fex and uxe on each side of the column Fx are supported by the pressure of the atmosphere on the surface of the water in the vessel MN; for if this vessel be placed in the exhausted receiver of an air-pump, these portions of water will not be sustained.

120. The preceding experiment completely overturns the hypothesis of Dr Hamilton and Dr Matthew Young, that the fluid was sustained in the tube by the lower ring of glass contiguous to the bottom of the tube, that this ring raises the portion of water immediately below it, and then other portions successively till the portion of water thus raised be in equilibrium with the attraction of the annulus in question.

121. Various experiments on the ascent of fluids in capillary tubes have been made by MM. Weitbrecht, Gelert, Lord Charles Cavendish, MM. Haüy and Tremery, Sir David Brewster, and M. Gay Lussac.

The following are the results obtained by M. Weitbrecht in the ascent of water:

| Diameter of the tube | Height of ascent | Constant quantity | |----------------------|-----------------|------------------| | In English inches | In inches | | | 0.065 | 0.72 | 0.0432 | | 0.045 | 0.95 | 0.0427 | | 0.03 | 0.53 | 0.0424 | | 0.025 | 1.72 | 0.043 |

Mean...........................................0.04282

122. The most accurate experiments on the depression of mercury in capillary tubes are those made by Lord Charles Cavendish:

| Interior diameter of tube | Mercury in one inch of tube | Depression of the mercury in inches | |---------------------------|-----------------------------|-----------------------------------| | 0.06 Inches | 972 grains | 0.005 inches | | 0.05 | 675 | 0.007 | | 0.04 | 432 | 0.015 | | 0.03 | 243 | 0.036 | | 0.02 | 108 | 0.067 | | 0.01 | 27 | 0.140 |

The constant quantity deduced by Dr Thomas Young from the preceding experiments is 0.015.

123. The very great discrepancy in the preceding results, obtained by very accurate and skilful observers, induced Sir David Brewster to repeat the experiments with an instrument constructed for the purpose, and to take such precautions, that he could always obtain the same results after repeated trials.

Having obtained a glass tube 7.9 inches long, and of a uniform circular bore, he took a wire of a less diameter than the bore of the tube, and formed a small hook at one of its ends. This hook was fastened to the middle of a worsted thread, of such a size as, when doubled, to fill the bore of the tube. The wire was then passed through the tube, and the worsted thread drawn after it; and when the whole was plunged in an alkaline solution, the worsted thread was fixed at one end, and the tube was drawn backwards and forwards till it was completely deprived, by its friction on the thread, of any grease or foreign matter which might have adhered to it. The tube and thread were then taken to clean water, and the same operation was repeated.

When the tube was thus perfectly cleaned, it was fixed vertically, by means of a level, in the axis of a piece of wood D (fig. 24), supported by the arm AD, fixed upon a stand AB; and it was also furnished with an index mn, which was moveable to and from the extremity b. On the arm CE, moveable in a vertical direction by the nut C, was placed a glass vessel F, containing the fluid, and nearly filled with it. The nut C was then turned till the extremity b of the tube touched the surface of the fluid, which was indicated by the sudden rise of the liquor round its sides. The fluid then rose in the tube till it remained stationary, and the index mn was moved till its extremity n pointed out the exact position of the upper surface of the fluid. In this situation, the distance ab was a measure of the ascent of the liquid above its level in the vessel F. In order to ascertain, however, whether the fluid was stationary, in consequence of any obstruction in the tube, or of an equilibrium of the attracting forces, the vessel with the fluid was raised a little higher than its former position, by means of the nut C, and then depressed below it. If the fluid now rose a little above n, and afterwards sunk a little below it, so as always to rise and fall with facility and uniformity along with the surface of the fluid in the vessel, it was obvious that it suffered no obstruction in the tube, and that ab was the accurate measure of its height. By separating the extremity b of the tube from the surface of the fluid, the fluid always rises above n; but upon again bringing them into contact, the fluid resumes its position at n. If there should be any portion of fluid at the end b of the tube, when it is again brought in contact with the fluid surface the water would rise around it before it had reached the In order to avoid this source of error, the index should have a projecting arm \( mn \) (fig. 25), carrying a screw \( st \), whose sharp point \( t \) can be easily brought on a level with the end \( b \) of the tube. When the extremity \( t \), therefore, which can always be kept dry, comes in contact with the fluid surface \( PQ \), the extremity \( b \) must also be exactly on the same level, even though the fluid had already risen around it. The tube was then cleaned, as formerly, for a subsequent observation. The results which were thus obtained for a great variety of fluids, and with a tube 0.0561 of an inch in diameter, are given in the following table:

| Names of Fluids | Height of ascent in inches | Constant quantity | |-----------------|---------------------------|------------------| | Water | 0.587 | 0.0327 | | Very hot water | 0.537 | 0.0301 | | Maritatic acid | 0.442 | 0.0248 | | Oil of boxwood | 0.427 | 0.0240 | | Oil of cassis | 0.420 | 0.0238 | | Nitrous acid | 0.413 | 0.0232 | | Oil of rapeseed | 0.404 | 0.0227 | | Castor oil | 0.403 | 0.0225 | | Nitrile acid | 0.395 | 0.0222 | | Oil of spermacti| 0.392 | 0.0220 | | Oil of almonds | 0.387 | 0.0217 | | Oil of olive | 0.387 | 0.0215 | | Balsam of Peru | 0.377 | 0.0212 | | Maritae of antimony | 0.373 | 0.0209 | | Oil of rhodium | 0.366 | 0.0205 | | Oil of pimento | 0.361 | 0.0203 | | Cajuput oil | 0.357 | 0.0200 | | Balsam of capivi| 0.357 | 0.0200 | | Oil of pennyroyal| 0.355 | 0.0199 | | Oil of thyme | 0.354 | 0.0199 | | Oil of bricks distilled from spermacti oil | 0.354 | 0.0199 | | Oil of caraway seeds | 0.353 | 0.0198 | | Oil of rue | 0.353 | 0.0198 | | Oil of rose | 0.351 | 0.0197 | | Balsam of sulphur | 0.349 | 0.0196 | | Oil of sweet fennel seeds | 0.349 | 0.0195 | | Oil of hyssop | 0.349 | 0.0195 | | Oil of rosemary | 0.344 | 0.0193 | | Oil of bergamot | 0.343 | 0.0192 | | Oil of amber | 0.343 | 0.0192 | | Oil of anise seeds | 0.342 | 0.0192 | | Oil of Barbadoes tar | 0.341 | 0.0191 | | Larkspur | 0.340 | 0.0191 | | Oil of cloves | 0.334 | 0.0187 | | Oil of turpentine | 0.333 | 0.0187 | | Oil of lemon | 0.333 | 0.0187 | | Oil of lavender | 0.328 | 0.0184 | | Oil of camomile | 0.327 | 0.0184 | | Oil of peppermint | 0.327 | 0.0184 | | Oil of sassafras | 0.327 | 0.0184 | | Highland whisky | 0.327 | 0.0184 | | Brandy | 0.326 | 0.0183 | | Oil of wormwood | 0.325 | 0.0183 | | Oil of dill seed | 0.324 | 0.0182 | | Oil of ambergris | 0.323 | 0.0181 | | German oil of juniper | 0.321 | 0.0180 | | Oil of nutmeg | 0.320 | 0.0180 | | Alcohol | 0.317 | 0.0178 | | Oil of savine | 0.310 | 0.0174 | | Ether | 0.285 | 0.0160 | | Oil of wine | 0.273 | 0.0153 | | Salphoric acid | 0.200 | 0.0112 |

124. By means of an instrument similar in principle to the one above described, M. Gay Lussac made a series of accurate experiments on the ascent of water and alcohol in capillary tubes. In these experiments the tubes were well wetted with the fluid.

The constant quantity in English inches, as deduced from these two experiments, is 0.04622.

Experiments with Alcohol.

| Diameter of the tube. | Height of ascent above lowest point of concavity. | Density of alcohol. | |-----------------------|---------------------------------------------------|---------------------| | 1.29441 millim. | 9.18235 millim. | 0.91961 | | 1.90381 | 6.08397 | 0.91961 | | 1.29441 | 9.30079 | 0.8595 | | 1.29441 | 9.99727 | 0.94153 | | 3.10-508 | 0.3835 | 0.81347 |

The temperature of the alcohol was 8°5 centigr., and the constant quantity for the two first experiments, reduced to English inches, is 0.01815, which agrees remarkably with 0.0178, the constant quantity in Sir David Brewster's experiments.

Experiments with Oil of Turpentine.

| Diameter of tube. | Height of fluid. | Density. | |-------------------|------------------|----------| | 1.29441 millim. | 9.25159 | 0.889458 |

This result also coincides very nearly with that of Sir David Brewster.

125. The following table contains a general view of the results obtained by different philosophers, from the ascent of water in capillary tubes.

| Names of observers. | Constant quantity, in English inches. | |--------------------|--------------------------------------| | Sir Isaac Newton | 0.020 | | MM. Haas and Tremery| 0.021 | | M. Carré, mean of three observations | 0.022 | | M. Hallstrom | 0.026 | | Sir David Brewster | 0.023 | | Munchenbroek | 0.039 | | M. Weithaupt, average of his results | 0.042 | | M. Gay Lussac, average of two observations | 0.048 | | Benjamin Martin | 0.048 | | Mr Atwood | 0.053 | | James Bernouilli | 0.064 |

Throwing aside the measure of James Bernouilli as obviously erroneous, we obtain 0.035 as the general average result of the preceding means; but the difference between this and the extreme measures of Newton and Atwood is so great, that there must be some cause, different from an error of observation, to which it is owing. The difference between the results obtained by Sir David Brewster and M. Gay Lussac, made with nice instruments founded on the same principle, leads to the same conclusion. Laplace indeed has ascribed, and we think justly, these differences to the greater or less degree of humidity on the sides of the tubes; and he informs us that Gay Lussac made his experiments with tubes very much wetted. Here, then, we have at once the cause of the difference above mentioned, because the experiments of Sir David Brewster were made with a tube carefully cleaned and dried after each experiment. A dry tube must necessarily raise the water to a less height than a wet one, and the difference must increase as the diameter of the tube employed is diminished. If we conceive a tube, indeed, with an exceedingly small bore, wetted over the whole of its interior, in the slightest degree, the two inner surfaces of the film would nearly meet in the axis, and the height of ascent would be infinite, or as high as the tube was long.

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1 Dr Young found the height of ascent of water and diluted spirit of wine to be as 100 to 64. 2 This is a mean of five experiments. 3 See Blot's Traité de Physique. 4 Opere, p. 366, 3d edition. 126. From these observations, the reader will be already prepared to draw the conclusion, that the ascent of fluids in glass tubes is a very equivocal measure of the force of capillary attraction, independent of its being applicable only to the single substance of glass.

With the view of removing this objection, Sir David Brewster long ago constructed an instrument, the object of which was to measure, upon an optical principle, the diameter of the circle of fluid which any cylindrical solid raises by capillary attraction above its general level. Thus, let MN (fig. 26) be the plan of a vessel filled with fluid, and A the section of a vertical cylinder of well cleaned and well dried glass, or any other substance not porous. This solid, A, will raise the fluid to a certain height around it, elevating a circular portion, CD, of the fluid above the general level; and it is manifest that the diameter CD of this elevated portion will be proportioned to the height of the fluid round the sides of the cylinder, or to the capillary force by which it is raised. In order to measure the diameter of this circle of fluid, a micrometer carries a small vertical frame along the edge FG of the vessel. Along this frame are stretched two fine parallel wires, whose images can be seen by reflection from the surface of the fluid, by an eye on the side PQ of the vessel, aided by a microscope with a distant focus. When the image of these wires is seen by reflection from any part of the fluid surface without the circle CD, it will suffer no change of form; but when it is seen by reflection from any portion of the elevated portion CD, the fibres will appear disturbed, and will indicate, by their return to the rectilinear form of accurate parallelism, the apparent termination of the circle CD. The same observation is made on the other side of A, at the boundary D, and a measure is thus obtained of the diameter of the circle CD, by means of the micrometer screw, by which the microscope on the side PQ, and the wire frame on the side FG, are moved (being fixed to the same frame) along the sides of the vessel. In this way solids of all kinds may be used, and their exterior or acting surfaces may be easily cleared from grease and other adhering substances.

This apparatus may be improved by using two cylinders, A, B, in place of one, and by moving one of them, suppose B, from the other, A, till the two elevated circular portions CD, DE disturb the images of the wires, seen by reflection from the intermediate point at D. A telescope may be advantageously used to observe the disturbance of any image seen reflected from the fluid surface.

127. When water is made to pass through a capillary tube of such a bore that the fluid is discharged only by successive drops, the tube, when electrified, will furnish a constant and accelerated stream, and the acceleration is proportional to the smallness of the bore. A similar effect may be produced by employing warm water. Sir John Leslie found that a jet of warm water rose to a much greater height than a jet of cold water, though the water in both cases moved through the same aperture, and was influenced by the same pressure. A siphon also, which discharged cold water only by drops, furnished warm water in an uninterrupted stream.

128. Such are the leading phenomena of capillary tubes. The rise of fluids between two plates of glass remains to be considered; and while it furnishes us with a very beautiful experiment, it confirms the reasoning by which we have accounted for the elevation of fluids in cylindrical canals. Let ABEF and CDEF (fig. 27) be two planes of plate glass with smooth and clean surfaces, having their sides EF joined together with wax, and their sides AB, CD kept at a little distance by another piece of wax W, so that their interior surfaces, whose common intersection is the line EF, may form a small angle. When this apparatus is immersed in a vessel, MN, full of water, the fluid will rise in such a manner between the glass planes as to form the curve DqomF, which represents the surface of the elevated water. By measuring the ordinates mn, op, &c., of this curve, and also its abscissae Ea, Ep, &c., Mr Hawksbee found it to be the common Apollonian hyperbola, having for its asymptotes the surface DE of the fluid, and EF the common intersection of the two planes. The following are the results which he obtained when the inclination of the planes was 20°:

| Distances from the touching ends of the plates | Heights of the water at the preceding distances | |-----------------------------------------------|-----------------------------------------------| | 13 Inches | 1 | | 9 | 2 | | 7 | 3 | | 5 | 5 | | 3 | 9 | | 2½ | 12 | | 1⅔ | 18 | | 1¼ | 27½ | | 1⅛ | 35 | | 1⅛ | 50 | | 1⅛ | 76 |

By repeating these observations at inclinations of 40°, and at various other angles, Mr Hawksbee found that the curve was an exact hyperbola in all directions of the planes. To the very same conclusion we are led by the principles already laid down; for as the distance between the plates diminishes at every point of the curve DqomF from D towards F, the water ought to rise higher at o than at q, still higher at m, and highest of all at F, where the distance between the plates is a minimum. To illustrate this more clearly, let ABEF and CDEF (fig. 28) be the same plates of glass (inclined at a greater angle for the sake of distinctness), and let FmDq, and FoqB be the curves which bound the surface of the elevated fluid. Then, since the altitudes of the water in capillary tubes are inversely as their diameters, or the distances of their opposite sides, the altitudes of the water between two glass plates should at any given point be inversely as the distances of the plates at that point. Now, the distance of the plates at the point m is obviously mo, or its equal mp, and the distance at q is qo or rq; and since mn is the altitude of the water at m, and qr its altitude at q, we have mn : qr = rq : mp; but (Geometry, Sect. IV.) Capillary Theor. xvii.) En : Er = mp : rt; therefore mn : qr = Er : En.

Attraction, that is, the altitudes of the fluid at the points m, q, which are equal to the abscissae En, Er (fig. 27), are proportional to the ordinates qr, mn, equal to the abscissae Er, En, in fig. 27. But in the Apollonian hyperbola the ordinates are inversely proportional to their respective abscissae; therefore the curve DqmF is the common hyperbola.

129. Mr Hawksbee extended his experiments to plates of glass placed parallel to each other, and separated to different distances, and he obtained the following results:

| Distance of plates | Height of ascent | Constant quantity | |-------------------|-----------------|------------------| | 0.0625 of an inch | 0.166 of an inch | 0.0104 | | 0.03125 | 0.333 | 0.0104 | | 0.015625 | 0.666 | 0.0104 | | 0.007802 | 1.333 | 0.0104 |

The following experiments on the same subject have been more recently made by M. Monge, MM. Hailly and Tremery, and M. Gay Lussac.

In those made by M. Monge, the plates were first cleaned with caustic alkali, and well washed. Their degree of separation was ascertained by silver wires of different thicknesses, and the fluid used was the filtered water of the Seine.

The following were the results:

| Distance of plates in parts of a line | Height of ascent | Constant quantity | |--------------------------------------|-----------------|------------------| | 0 or 0.0101 inch | 15.5 lines | 0.1555 | | 0.0068 | 33.5 | 0.2278 | | 0.0030 | 74 | 0.222 |

The following result was obtained by MM. Hailly and Tremery:

| Distance of plate | Height of ascent | Constant quantity | |-------------------|-----------------|------------------| | 1 millimetre | 6.5 millimetres | 6.5 |

The following measures were obtained by M. Gay Lussac, with plates of glass ground perfectly flat:

| Distance of plates | Height of ascent above lowest point of concavity | Temp. centigr. | |-------------------|---------------------------------------------------|---------------| | 1.069 millimetre | 13.574 | 16° |

Here the constant quantity is 14.51, or 0.02251, when reduced to English inches for a distance of \( \frac{1}{18} \)th of an inch.

130. The phenomena which we have been considering are all referrible to one simple fact, that the particles of glass have a stronger attraction for the particles of water than the particles of water have for each other. This is the case with almost all other fluids except mercury, the particles of which have a stronger attraction for each other than for glass. When capillary tubes, therefore, are plunged in this fluid, a new series of phenomena present themselves to our consideration. Let MN (fig. 29) be a vessel full of mercury. Plunge into the fluid the capillary tube CD, and the mercury, instead of rising in the tube, will remain stationary at E, its depression below the level surface AB being inversely proportional to the diameter of the bore. This was formerly ascribed to a repulsive force supposed to exist between mercury and glass, but we shall presently see that it is owing to a very different cause.

131. That the particles of mercury have a very strong attraction for each other, appears from the globular form which a small portion of that fluid assumes, and from the resistance which it opposes to any separation of its parts. If a quantity of mercury is separated into a number of minute parts, all these parts will be spherical; and if two of these spheres be brought into contact, they will instantly rush together, and form a single drop of the same form. There is also a very small degree of attraction existing between glass and mercury; for a globule of the latter very readily adheres to the lower surface of a plate of glass.

Now, suppose a drop of water laid upon a surface anointed with grease, to prevent the attraction of cohesion from reducing it to a film of fluid, this drop, if very small, will be spherical. If its size is considerable, the gravity of its parts will make it spheroidal, and as the drop increases in magnitude, it will become more and more flattened at its poles, like AB in fig. 30. The drop, however, will still retain its convexity at the circumference, however oblate be the spheroid into which it is moulded by the force of gravity. Let two pieces of glass oAm, pBn be now brought in contact with the circumference of the drop; the mutual attraction between the particles of water which enabled it to preserve the convexity of its circumference, will yield to their superior attraction for glass; the space maap will be immediately filled; and the water will rise on the sides of the glass, and the drop will have the appearance of AB in fig. 31. If the drop AB (fig. 30) be now supposed mercury instead of water, it will also, by the gravity of its parts, assume the form of an oblate spheroid; A but when the pieces of glass oAm, pBn are brought close to its periphery, their attractive force upon the mercurial particles is not sufficient to counteract the mutual attraction of these particles; the mercury, therefore, retains its convexity at the circumference, and assumes the form exhibited in fig. 32. The small spaces o, p (fig. 30) being filled by the pressure of the superincumbent fluid, while the spaces below m, n still remain between the glass and the mercury. Now if the two plates of glass A, B be made to approach each other, the depressions m, n will still continue, and when the distance of the plates is so small that these depressions or indentations meet, the mercury will sink between the plates, and its descent will continue as the pieces of glass approach. Hence the depression of the mercury in capillary tubes becomes very intelligible. If two glass planes forming a small angle, as in fig. 27, be immersed in a vessel of mercury, the fluid will sink below the surface of the mercury in the vessel, and form an Apollonian hyperbola like DoF, having for its asymptotes the common intersection of the planes and the surface of mercury in the vessel.

132. The depression of mercury in capillary tubes is evidently owing to the greater attraction which the particles of mercury have for each other than for glass. The difference between these two attractions, however, arises from an imperfect contact between the mercury and the capillary tube, occasioned by the interposition of a thin coating of water which generally lines the interior surface of the tube, and weakens the mutual action of the glass and mercury; for this action always increases as the thickness of the interposed film is diminished by boiling. In the experiments which were made by Laplace and Lavoisier on barometers, by boiling the mercury in them for a long time, the convexity of the interior surface of the mercury was often made to disappear. They even succeeded in rendering it concave, but could always restore the convexity by introducing a drop of water into the tube. When the ebullition of the mercury is sufficiently strong to expel all foreign particles, it often rises to the level of the surrounding fluid, and the depression is even converted into an elevation.

133. Between mercury and water there is likely to be some fluid in which the attraction of the glass for its particles is nearly equal to half the attraction of the fluid for itself. Sir David Brewster has observed that iodine dissolved in chloride of sulphur approximates to this condition; but not having the chloride by itself, he could not observe whether or not the effect is produced or influenced by the iodine. If it is, then a solution may be obtained in which the above condition is perfect. The solution of the iodine already mentioned scarcely rises on the sides of the glass ball which contains it.

134. As most philosophers seem to agree in thinking that all the capillary phenomena are referable to the cohesive attraction of the superficial particles only of the fluid, a variety of experiments has been made in order to determine the force required to raise a horizontal solid surface from the surface of a fluid. Mr Achard found that a disc of glass, 1½ French inches in diameter, required a weight of 91 French grains to raise it from the surface of the water at 69° of Fahrenheit, which is only 37 English grains for each square inch. At 444° of Fahrenheit the force was ¼ th greater, or 39½ grains; the difference being ¼ th for each degree of Fahrenheit. From these experiments Dr Young concludes that the height of ascent in a tube of a given bore, which varies in the duplicate ratio of the height of adhesion, is diminished about ¼ th for every degree of Fahrenheit that the temperature is raised above 50°; and he conjectures that there must have been some considerable source of error in Achard's experiments, as he never found this diminution to exceed ¼ th.

135. According to the experiments of Morveau, the force necessary to elevate a circular inch of gold from the surface of mercury is 446 grains; a circular inch of silver, 429 grains; a circular inch of tin, 418 grains; a circular inch of lead, 397 grains; a circular inch of bismuth, 372 grains; a circular inch of zinc, 204 grains; a circular inch of copper, 142 grains; a circular inch of metallic antimony, 126 grains; a circular inch of iron, 115 grains; and a similar surface of cobalt required 8 grains. The order in which these metals are arranged is the very order in which they are most easily amalgamated with mercury.

The most recent experiments on the adhesion of surfaces to fluids have been made by M. Gay Lussac, who obtained the following results with a circular plate of glass 118-396 millimetres in diameter:

| Name of fluid | Weight necessary to raise the plate from the glass | Specific gravity | |---------------|---------------------------------------------------|----------------| | Water | 59-40 grammes | 1-000 | | Alcohol | 31-08 | 0-8196 | | Alcohol | 32-87 | 0-8505 | | Oil of turpentine | 37-152 | 0-9415 |

With a copper disc, 116-604 millimetres in diameter, the weight necessary to raise it from water, at the temperature of 18°-5 centigrade, was 57-945 grammes, differing very little, if at all, from glass; for the diminution of weight may be explained by the circumstance of the copper disc being nearly two millimetres less in diameter than the glass. In these experiments the discs were suspended from the scale of a balance, and the weights in the other scale successively increased till the force of adhesion was overcome at the instant when the disc detached itself from the fluid surface.

136. The approach of two floating bodies has been ascribed by some to their mutual attraction, and by others to the attraction of the portions of fluid that are raised round each by the attraction of cohesion. Dr Young, however, observes, that the approach of the two floating bodies is produced by the excess of the atmospheric pressure on the remote sides of the solids, above its pressure on their neighbouring sides; or, if the experiments are performed in a vacuum, by the equivalent hydrostatic pressure or suction derived from the weight and immediate cohesion of the intervening fluid. This force varies alternately in the inverse ratio of the square of the distance; for when the two bodies approach each other, the altitude of the fluid between them is increased in the simple inverse ratio of the distance; and the mean action, or the negative pressure of the fluid on each particle of the surface, is also increased in the same ratio. When the floating bodies are surrounded by a depression, the same law prevails, and its demonstration is still more simple and obvious.

137. A different view of the subject has been given by Monge, who made a number of accurate experiments on the subject, and deduced from them the following laws:

1st. If two floating bodies, capable of being wetted with the fluid on which they float, are placed near each other, they will approach as if mutually attracted.

In order to explain this law, let AB, CD (fig. 33) be two suspended plates of glass, placed at such distance that the point H, where the two portions of elevated fluid meet, is on a level with the rest of the water, the two plates will remain stationary and in perfect equilibrium. But if they are brought nearer one another, as in fig. 34, the water will rise between them to a point H above the level, and by a nearer approximation, to the point G. The water thus elevated, acting like a curved chain hung to the two plates, attracts the sides of the plates, and brings them together in a horizontal direction. The very same thing takes place with the floating bodies A, B, placed at such a distance that the water rises between them above its level, and hence these bodies will approach by the attraction of the fluid on their inner sides.

2nd. When the two floating bodies A, B (fig. 35), are not capable of being wetted, they will approach each other as if mutually attracted, when they are placed near one another.

In this case the fluid is depressed between them below its natural level H, and the two bodies are pressed inwards or towards each other, which pressure being greater than the pressure outwards of the fluid between them, they will approach each other by the action of the difference of these pressures.

3rd. If one of the bodies, A, is capable of being wetted, and the other, B, not, as shown in the fig. 37, they will recede from each other as if mutually repelled.

As the fluid rises round A, and is depressed round B, the depression round B will not be equal all round, and hence the body B, being placed as if on an inclined plane, will move to the right hand where the pressure is the least.

In this last case, Laplace was led by theory to believe Capillary that when B is placed very near A, the repulsion will be converted into attraction.

M. Haüy tried this experiment with planes of ivory and talc, the former being incapable of being wetted with water and the latter not; and he found, in conformity with Laplace's prediction, that at a certain short distance the tale moved suddenly into contact with the ivory.

138. The phenomena of attraction and repulsion exhibited between small lighted wicks, swimming in a basin of oil, and the motions of floating evaporable substances like camphor, and also of potassium and light substances, such as cork, impregnated with ether, have been sometimes treated under this head. The first of these classes of phenomena arise from an unbalanced pressure upon the floating wick, arising from a difference of temperature of different parts of the oil; and the movements of the second class arise from the reaction of the currents of vapour which flow from the floating substances. A full account of these phenomena will be found in the *Edinburgh Transactions* (vol. iv., p. 44), in the *Mémoires présentés à l'Institut* (tom. i., p. 125), and the more recent observations of Matteucci, in the *Annales de Chimie* (June 1833, tom. liii., p. 216-219).

**Theory of Capillary Attraction.**

139. Dr Hook was one of the earliest speculators in capillary attraction. He believed that the phenomena were due to a diminution of pressure within the tube by reason of friction against its inner surface. Hawksbee, by means of the air-pump, showed that this was erroneous. Hawksbee was the first who made an approximation to the true theory of capillarity, by ascribing it to the attraction of the tube or plate. Dr Jurin corrected one of Hawksbee's experiments, together with its explanation. Newton also seems to have been in some measure acquainted with capillary attraction, if we may judge from the 31st query of the last edition of his *Optics*. But before Clairaut took up the subject, the prevalent theories on capillarity were defective in two respects; they contained no calculation founded on the hypothesis of an attraction, sensible only at insensible distances from the attracting centres, although Newton had then shown the existence of such forces; and no account of the cohesive attraction of the parts of the fluid for each other were recognised. Clairaut, then, was the first to see the necessity of taking into account the action of the fluid on itself; he was the first mathematician who attempted to analyze the forces which contribute to the ascent of fluids in capillary tubes. After pointing out the insufficiency of preceding theories, he gives an analysis of the different forces which contribute to the suspension of fluids in capillary tubes.

Let ABCDEFGH (fig. 38) be the section of a capillary tube, MNP the surface of the water in the vessel, Ii the height of its ascent, viz., the concave surface of the fluid column, and IKLM an indefinitely small column of fluid reaching to the surface at M. Now the column ML is solicited by the force of gravity which acts through the whole extent of the column, and by the reciprocal attraction of the molecules which, though they act the same in all the points of the column, only exhibit their effects towards the extremity M. If any particle e is taken at a less distance from the surface than the distance at which the attraction of the liquid generally terminates, and if mn is a plane parallel to MN, and at the same distance from the particle e, then this particle will be equally attracted by the water between the planes MN, mn. The water, however, below mn, will attract the particle downwards, and this effect will take place as far as the distance where the attraction ceases.

The formula obtained by Clairaut for the altitude ii (fig. 38) is

$$Ii = \frac{(2Q - Q')}{p} \int dx [b, x] + \int dx [b, x, Q, Q']$$

in which Q is the intensity of the attraction of the glass, Q' the intensity of the attraction of the water, b the interior radius of the tube, and p the force of gravity.

Clairaut then observes, that there is an infinitude of pos-

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1 *Théorie de la Figure de la Terre*, chap. x. Paris, 1743, 1808. Capillary sible laws of attraction which will give a sensible quantity of attraction, for the elevation of the fluid If above the level MN, when the diameter of the tube is very small, and a quantity next to nothing when the diameter is considerable; and he remarks, that we may select the law which gives the inverse ratio between the diameter of the tube and the height of the liquid, conformable to experiment.

140. It follows from the preceding formula, that if any solid, AB (fig. 39), possesses half the attracting power of the fluid CD, the surface of the fluid will remain horizontal; for the attraction being represented by DA, DE, and DC, DA and DE may be combined into DB, and DB and DC into DF, which is vertical. The water will therefore not be raised, since the surface of a fluid at rest must be perpendicular to the resulting direction of all the forces which act upon it.

When the attracting power of the solid is more than half as great, the resultant of the forces will be GF (fig. 40), and therefore the fluid must rise towards the solid, in order to be perpendicular to GF. When the attractive power of the solid is less than that of the fluid, the resultant will be HF; and therefore, as in the case of mercury, the surface must be depressed, in order to be perpendicular to the force.

141. In 1751, Segner, without having seen any of Clai-Dr Young's termination of the form of the surface of a water-drop resting on a horizontal plane, assuming that the particles of the fluid have an attraction for each other. The nature of this problem is the same as that of determining the form of the upper surface of a fluid column sustained in a capillary tube: neither of these problems had engaged the attention of mathematicians. Segner proposed a theory, but he owned that it was defective.

Monge, in 1787, attempted an explanation of the apparent attraction and repulsion of small bodies floating on fluids: the principle on which he grounds his explanations is the same as the more exact interpretations of the same phenomenon by Young, Laplace, and Poisson. Monge did not follow up his investigations by analysis.

Dr Young's Essay on the Cohesion of Fluids, containing his theory of capillary attraction, was read before the Royal Society, December 1804. His views of the subject are similar to those of Segner and Monge. He refers the phenomena observed in capillary tubes to the cohesive attraction of the superficial particles of the fluid, in so far as it gives rise to a uniform tension of the surface. In support of his theory, he makes two assumptions—1st, that the tension of the fluid surface is known; 2d, that at the juncture of a fluid surface with the surface of a solid, there is an appropriate angle of contact between the two surfaces. This angle for glass and water is nearly evanescent, whereas for glass and mercury it is about 140°; on these two assumptions, he thinks that a theory of capillary attraction can be satisfactorily built. In the latter part of his essay, he shows how his assumptions may be derived from ulterior physical principles.

The subject of capillary attraction has more recently occupied the attention of the Marquis de Laplace, who published his theory in 1806. In 1807 he published a supplement to his theory, in which he compares his formula with the experiments of Gay Lussac and others.

In the first treatise published by M. Laplace, his method of considering the phenomena was founded on the form of the surface of the fluid in capillary spaces, and on the conditions of equilibrium of this fluid in an infinitely narrow canal, resting by one of its extremities upon this surface, and by the other on the horizontal surfaces of an indefinite fluid, in which the capillary tube was immersed. In his supplement to that treatise, he has examined the subject in a much more popular point of view, by considering directly the forces which elevate and depress the fluid in this space. By this means, he is conducted easily to several general results, which it would have been difficult to obtain directly by his former method.

142. Let AB (fig. 41) be a vertical tube whose sides are perpendicular to its base, and which is immersed in a fluid that rises in the interior of the tube above its natural level. A thin film of fluid is first raised by the action of the sides of the tube; this film raises a second film, and this second film a third film, till the weight of the volume of fluid raised exactly balances all the forces by which it is actuated. Hence it is obvious, that the elevation of the column is produced by the attraction of the tube upon the fluid, and the attraction of the fluid for itself. Let us suppose that the inner surface of the tube AB is prolonged to E, and after bending itself horizontally in the direction ED, that it assumes a vertical direction DC; and let us suppose the sides of this tube to be so extremely thin, or to be formed of a film of ice, so as to have no action on the fluid which it contains, and not to prevent the reciprocal action which takes place between the particles of the first tube AB and the particles of the fluid. Now, since the fluid in the tubes AE, CD is in equilibrium, it is obvious that the excess of pressure of the fluid in AE is destroyed by the vertical attraction of the tube; and of the fluid upon the fluid contained in AB. In analyzing these different attractions, Laplace considered first those which take place under the tube AB. The fluid column BE is attracted, 1st, by itself; 2d, by the fluid surrounding the tube BE. But these two attractions are destroyed by the similar attractions experienced by the fluid contained in the branch DC, so that they may be entirely neglected. The fluid in BE is also attracted vertically by the fluid in AB; but this attraction is destroyed by the attraction which it exercises in the opposite direction upon the fluid in BE, so that these balanced attractions may likewise be neglected. The fluid in BE is likewise attracted vertically upwards by the tube AB, with a force which we shall call Q, and which contributes to destroy the excess of pressure exerted upon it by the column BF raised in the tube above its natural level.

Now, the fluid in the lower part of the round tube AB is attracted, 1. By itself; but as the reciprocal attractions of a body do not communicate to it any motion if it is solid, we may, without disturbing the equilibrium, conceive the fluid in AB frozen. 2. The fluid in the lower part of AB is attracted by the interior fluid of the tube BE, but as the latter is attracted upwards by the same force, these two actions may be neglected as balancing each other. 3. The fluid in the lower part of BE is attracted by the fluid which surrounds the ideal tube BE, and the result of this attraction is a vertical force acting downwards, which we may call -Q, the contrary sign being applied, as the force is here opposite to the other force Q. As it is highly probable that the attractive forces exercised by the glass and the water vary according to the same function of the distance, so as to differ only in their intensities, we may employ the constant coefficients \( \rho \) as measures of their intensity, so that the forces Q and -Q' will be proportional to \( \rho \) and \( \rho' \); for the interior surface of the fluid which surrounds the tube... Capillary BE, is the same as the interior surface of the tube AB. Consequently, the two masses, viz., the glass in AB, and the fluid round BE, differ only in their thickness; but as the attraction of both these masses is insensible at sensible distances, the difference of their thicknesses, provided their thicknesses be sensible, will produce no difference in the attractions.

4. The fluid in the tube AB is also acted upon by another force, namely, by the sides of the tube AB in which it is inclosed. If we conceive the column FB divided into an infinite number of elementary vertical columns, and if at the upper extremity of one of these columns we draw a horizontal plane, the portion of the tube comprehended between the plane and the level surface BC of the fluid will not produce any vertical force upon the column; consequently, the only active vertical force is that which is produced by the ring of the tube immediately above the horizontal plane. Now, the vertical attraction of this part of the tube upon BE, will be equal to that of the entire tube upon the column BE, which is equal in diameter, and similarly placed. This new force will therefore be represented by \(+Q\). In combining these different forces, it is manifest that the fluid column BF is attracted upwards by the two forces \(+Q, +Q\), and downwards by the force \(-Q\); consequently the force with which it is raised upwards will be \(2Q - Q'\).

If the force \(2Q\) is less than \(Q'\), then \(V\) will be negative, and the fluid will sink in the tube; but as long as \(2Q\) is greater than \(Q'\), \(V\) will be positive, and the fluid will rise above its natural level; as was long before shown by M. Clairaut.

Since the attractive forces, both of the glass and the fluid, are insensible at sensible distances, the surface of the tube AB will act sensibly only on the column of fluid immediately in contact with it. We may therefore neglect the consideration of the curvature, and consider the inner surface as developed upon a plane. The force \(Q\) will therefore be proportional to the width of this plane, or what is the same thing, to the interior circumference of the tube. Calling \(c\), the circumference of the tube, we shall have \(Q = \rho c\); \(\rho\) being a constant quantity, representing the intensity of the attraction of the tube AB upon the fluid, in the case where the attractions of different bodies are expressed by the same function of the distance. In every case, however, \(\rho\) expresses a quantity dependent on the attraction of the matter of the tube, and independent of its figure and magnitude. In like manner we shall have \(Q = \rho c\); \(\rho\) expressing the same thing with regard to the attraction of the fluid for itself; that \(\rho\) expressed with regard to the attraction of the tube for the fluid. By substituting these values of \(Q, Q'\), in the preceding equation, we have

\[ gDV = c(2\rho - \rho') \]

If we now substitute, in this general formula, the value of \(c\) in terms of the radius if it is a capillary tube, or in terms of the sides if the section is a rectangle, and the value of \(V\) in terms of the radius and altitude of the fluid column, we shall obtain an equation by which the heights of ascent may be calculated for tubes of all diameters, after the height, belonging to any given diameter, has been ascertained by direct experiment.

143. In the case of a cylindrical tube, let \(r\) represent the ratio of the circumference to the diameter, \(h\) the height of the fluid-column reckoned from the lower point of the meniscus, \(q\) the mean height to which the fluid rises, or the height at which the fluid would stand if the meniscus were to fall down and assume a level surface, then we have \(\pi r^2\) for the solid content of a cylinder of the same height and radius as the meniscus; and as the meniscus, added to the solid contents of the hemisphere of the same radius, must be equal to \(\pi r^2\), we have \(\pi r^2 = \frac{2\pi r^3}{3}\), or \(\frac{\pi r^3}{3}\), for the solid content of the meniscus. But since \(\frac{\pi r^3}{3} = \pi r^2 \times \frac{r}{3}\), it follows that the meniscus \(\frac{\pi r^3}{3}\) is equal to a cylinder whose base is \(\pi r^2\), and altitude \(\frac{r}{3}\). Hence, we have

\[ q = h + \frac{r}{3} \]

or what is the same thing, the mean altitude \(q\) in a cylinder is always equal to the altitude \(h\) of the lower point of the concavity of the meniscus increased by one-third of the radius, or one-sixth of the diameter of the capillary tube.

Now, since the contour \(c\) of the tube \(= 2\pi r\), and since the volume \(V\) of water raised is equal to \(q \times \pi r^2\), we have, by substituting these values in the general formula,

\[ gDq\pi r^2 = 2\pi r (2\rho - \rho') \] (No. 1)

and dividing by \(\pi r\) and \(gD\), we have,

\[ rq = 2 \frac{2\rho - \rho'}{gD} \quad \text{and} \quad q = 2 \frac{2\rho - \rho'}{gD} \times \frac{1}{r} \] (No. 2)

144. In applying this formula to Gay Lussac's experiments, we have the constant quantity,

\[ \frac{2\rho - \rho'}{gD} = rq = 647205 \times 281634 + 0215735 = 151311 \]

for Gay Lussac's first experiment. In order to find the height of the fluid in his second tube by means of this constant quantity, we have

\[ \frac{2\rho - \rho'}{gD} = rq = 647205 \times 281634 + 0215735 = 151311 \]

\[ = 158956, \text{from which, if we subtract one-sixth of the diameter, or } 03173, \text{ we have } 155783 \text{ for the altitude } h \text{ of the lower point of the concavity of the meniscus, which differs only } 00078 \text{ from } 15861, \text{ the observed altitude.} \]

If we apply the same formula to Gay Lussac's experiments on alcohol, we shall find the constant quantity

\[ \frac{2\rho - \rho'}{gD} = 60825 \text{ as deduced from the first experiment, } \]

and \(h = 60725\), which differs only by \(00100\) from \(608397\), the altitude observed.

From these comparisons, it is obvious that the mean altitudes, or the values of \(q\), are very nearly reciprocally proportional to the diameters of the tubes; for, in the experiments on water, the value of \(q\) deduced from this ratio is \(15993\), which differs little from \(159034\), the value found from experiment; and that, in accurate experiments, the correction made by the addition of the sixth part of the diameter of the tube is indispensably requisite.

145. If the section of the pipe in which the fluid ascends to rectilinear, whose greater side is \(a\), and its lesser side \(d\), then the base of the elevated column will be \(ad\), and spaces. Its perimeter \(c = 2a + 2d\). Hence, the value of the meniscus will be

\[ ad^2 - \frac{axd^2}{8} = \frac{ad^2}{2} \left(1 - \frac{\pi}{4}\right), \text{ that is } q = h + \frac{d}{2} \left(1 - \frac{\pi}{4}\right) \]

Hence, if in the general equation No. 1 we substitute for \(c\) its equal \(2a + 2d\), and for \(V\) its equal \(adq\), we have

\[ gDqad = 2\rho - \rho' \times 2a + 2d \] In applying this formula to the elevation of water between two glass plates, the side \(a\) is very great compared with \(d\), and therefore the quantity \(\frac{d}{a}\) being almost insensible, may be safely neglected. Hence the formula becomes

\[ q = 2 \frac{2\rho - \rho'}{gD} \times \frac{1}{d}. \]

By comparing this formula with the formula No. 2, it is obvious that water will rise to the same height between plates of glass as in a tube, provided the distance \(d\) between the two plates of glass is equal to \(r\), or half the diameter of the tube. This result was obtained by Newton, and has been confirmed by the experiments of succeeding writers.

As the constant quantity \(2 \frac{2\rho - \rho'}{gD}\) is the same as already found for capillary tubes, we may take its value, viz., 15·1311, and substitute it in the preceding equation, we then have

\[ q = \frac{15·1311}{1·060} = 14·1544; \text{ and since} \]

\[ h = q - \frac{d}{2} \left(1 - \frac{\pi}{2}\right), \text{ subtracting} \]

\[ \frac{d}{2} \left(1 - \frac{\pi}{4}\right) = 0·1147, \text{ we have} \]

\[ h = 14·0397, \text{ which differs very little from } 13·574, \text{ the observed altitude, and } d = 1·060. \]

It will be seen from the formula No. 2, that of all tubes that have a prismatic form, the hollow cylinder is the one in which the volume of fluid raised is the least possible, as it has the smallest perimeter. It appears, also, that if the section of the tube is a regular polygon, the altitudes of the fluid will be reciprocally proportional to the homologous lines of the similar base, a result which, as we have seen, M. Gellert obtained from direct experiment. Hence, in all prismatic tubes whose sections are polygons inscribed in the same circle, the fluid will rise to the same mean height. If one of the two bases is, for example, a square, and the other an equilateral triangle, the altitudes will be as 2 : 3, or very nearly as 7 : 8.

M. Laplace has remarked that there may be several states of equilibrium in the same tube, provided its width is not uniform. If we suppose two capillary tubes communicating with one another, so that the smallest is placed above the greatest, we may then conceive their diameters and lengths to be such that the fluid is at first in equilibrium above its level in the widest tube, and that in pouring in some of the same fluid, so as to reach the smaller tube, and fill part of it, the fluid will still maintain itself in equilibrium. When the diameter of a capillary tube diminishes by insensible gradations, the different states of equilibrium are alternately stable and unstable. At first the fluid tends to raise itself in the tube, and this tendency diminishing, becomes nothing in a state of equilibrium. Beyond this it becomes negative, and consequently the fluid tends to descend. Thus the first equilibrium is stable, since the fluid, being a little removed from this state, tends to return to it. In continuing to raise the fluid, its tendency to descend diminishes, and becomes nothing in the second state of equilibrium. Beyond this it becomes positive, and the fluid tends to rise, and consequently to remove from this state which is not stable. In a similar manner it will be seen, that the third state is stable, the fourth unstable, and so on.

Although the preceding method of considering the phenomena of capillary attraction is extremely simple and accurate, yet it does not indicate the connexion which subsists between the elevation and depression of the fluid, and the concavity or convexity of the surface which every fluid tore assumes in capillary spaces. The object of M. Laplace's first method, contained in his first supplement, is to determine this connexion.

By means of the methods for calculating the attraction of spheroids, he determines the action of a mass of fluid terminated by a spherical surface, concave or convex, upon a column of fluid contained in an infinitely narrow canal, directed towards the centre of this surface. By this action Laplace means the pressure which the fluid contained in the canal would exercise, in virtue of the attraction of its entire mass upon a plane base situated in the interior of the canal, and perpendicular to its sides, at any sensible distance from the surface, this base being taken for unity. He then shows that this action is smaller when the surface is concave than when it is plane, and greater when the surface is convex. The analytical expression of this action is composed of two terms. The first of these terms, which is much greater than the second, expresses the action of the mass terminated by a plane surface; and the second term expresses the part of the action due to the sphericity of the surface, or, in other words, the action of the meniscus comprehended between this surface and the plane which touches it. This action is either additive to the preceding, or subtractive from it, according as the surface is convex or concave. It is reciprocally proportional to the radius of the spherical surface; for the smaller that this radius is, the meniscus is the nearer to the point of contact.

From these results relative to bodies terminated by sensible segments of a spherical surface, Laplace deduces this general theorem: "In all the laws which render the attraction insensible at sensible distances, the action of a body terminated by a curve surface, upon an interior canal infinitely narrow, perpendicular to this surface in any point, is equal to half the sum of the actions upon the same canal of two spheres, which have for their radii the greatest and the smallest of the radii of the osculating circle of the surface at this point."

By means of this theorem, and the laws of hydrostatics, Laplace has determined the figure which a mass of fluid ought to take when acted upon by gravity, or contained in a vessel of a given figure. The nature of the surface is expressed by an equation of partial differences of the second order, which cannot be integrated by any known method. If the figure of the surface is one of revolution, the equation is reduced to one of ordinary differences, and is capable of being integrated by approximation, when the surface is very small. Laplace next shows, that a very narrow tube approaches the more to that of a spherical segment as the diameter of the tube becomes smaller. If these segments are similar in different tubes of the same substance, the radii of their surfaces will be inversely as the diameter of the tubes. This similarity of the spherical segments will appear evident, if we consider that the distance at which the action of the tube ceases to be sensible is imperceptible; so that if, by means of a very powerful microscope, this distance should be found equal to a millimetre, it is probable that the same magnifying power would give to the diameter of the tube an apparent diameter of several metres. The surface of the tube may therefore be considered as very nearly plane, in a radius equal to that of the sphere of sensible activity; the fluid in this interval will therefore descend, or rise from this surface very Capillary nearly as if it were plane. Beyond this, the fluid being subjected only to the action of gravity and the mutual action of its own particles, the surface will be very nearly that of a spherical segment of which the extreme planes, being those of the fluid surface, at the limits of the sphere of the sensible activity of the tube, will be very nearly in different tubes equally inclined to their sides. Hence it follows that all the segments will be similar.

149. The approximation of these results gives the true cause of the ascent or descent of fluids in capillary tubes in the inverse ratio of their diameter. If in the axis of a glass tube we conceive a canal infinitely narrow, which bends round like the tube ABEDC in fig. 41, the action of the water in the tube in this narrow canal will be less, on account of the concavity of its surface, than the action of the water in the vessel on the same canal. The fluid will therefore rise in the tube to compensate for this difference of action; and as the concavity is inversely proportional to the diameter of the tube, the height of the fluid will be also inversely proportional to that diameter. If the surface of the interior fluid is convex, which is the case with mercury in a glass tube, the action of this fluid on the canal will be greater than that of the fluid in the vessel, and therefore the fluid will descend in the tube in the ratio of their difference, and consequently in the inverse ratio of the diameter of the tube.

In this manner of viewing the subject, the attraction of capillary tubes has no influence upon the ascent or depression of the fluids which they contain, but in determining the inclination of the first planes of the surface of the interior fluid extremely near the sides of the tube, and upon this inclination depends the concavity or convexity of the surface, and the length of its radius. The friction of the fluid against the sides of the tube may augment or diminish a little the curvature of its surface, of which we see frequent examples in the barometer. In this case the capillary effects will increase or diminish in the same ratio.

The differential equation of the surfaces of fluids inclosed in capillary spaces of revolution, conduces Laplace to the following general result: That if into a cylindrical tube we introduce a cylinder which has the same axis as that of the tube, and which is such that the space comprehended between its surface and the interior surface of the tube has very little width, the fluid will rise in this space to the same height as in a tube whose radius is equal to this width. If we suppose the radii of the tube and of the cylinder infinite, we have the case of a tube included between two parallel and vertical planes, very near each other. This result has been confirmed, as we have already seen, by the experiments of Newton, Hailly, and Gay Lussac. Laplace then applies his theory to the phenomena presented by a drop of fluid, either in motion or suspended in equilibrium either in a conical capillary tube, or between two plates, and inclined to each other, as discovered by Mr Hawksbee; to the mutual approximation of two parallel and vertical discs immersed in a fluid; to the phenomena which take place when two plates of glass are inclined to each other at a small angle; and to the determination of the figure of a large drop of mercury laid upon a horizontal plate of glass.

150. Laplace has failed in giving a satisfactory proof of the constancy of the angle of contact, or that angle which the free surface of the fluid makes with its surface of contact. Gauss, in 1830, endeavoured to rectify the defect by forming his equations of equilibrium on the principle of virtual velocities (see Mechanics) which he applies to the whole mass of fluid, and not to a differential element, as Lagrange has done. By this method he obtains a sextuple integral, which extends to the whole mass, and which is to be a minimum; if the fluid be supposed homogeneous and incompressible, the integral becomes quadruple. Further, the integral may become single, if the only forces acting be gravity and the molecular attractions of the fluid and containing solid, and if the sphere of activity of the two attractions is insensible. He also arrived at two equations, one relative to the free surface, and which is Laplace's fundamental equation; the other relative to the angle of contact, and which corresponded with Dr Young's result.

151. Poisson's first memoir on the theory of capillary attraction was read before the Paris Academy, November 1828. In it his object is to form the equations of equilibrium of fluids on physical principles, i.e., by assuming that a mass of fluid is made up of distinct molecules, separated one from another by spaces excessively small, and void of ponderable matter.

As a preliminary to the discussion, the magnitude of these molecules, as also the spaces between them, are assumed to be so small, that a line which may be supposed to be a great multiple of them is of insensible magnitude. The molecules are attractive of each other, and at the same time are repulsive, owing to their proper heat; their actions and reaction are equal; the force decreases rapidly as the distance increases, and it is only sensible at insensible distances. The influence of these attractive centres is felt at distances very great when compared with the molecular spaces, and the rapid decrease to commence only at distances which are large multiples of these small intervals.

By molecular action is to be understood the excess of the repulsion over the attraction of the molecules, and this force is supposed to be different for different points of any two molecules. The mean value is called the principal force, and the secondary force is the variation from this normal value, according as different points of the molecules are directed towards each other. The secondary force is important in solids, insomuch as it gives rise to their rigidity and resistance to the lateral motion of their molecules; the want of this force in fluids causes their particles to be very mobile. It is thus that a fluid is distinguished by Poisson from a solid: If a point be taken anywhere in the interior of a fluid mass, and a straight line of insensible length, but a great multiple of the mean intervals between the molecules, be drawn in any direction from that point, the mean interval between the molecules that lie in the line is constant, though the particles may be irregularly disposed along it.

Reasoning on the ground of these suppositions, Poisson forms equations relative to the pressure in the interior of a fluid mass, which are the same as if he had started from the supposition of the equality of pressure in every direction. His reasoning on the above suppositions leads him to the explanation of a known fact; but for all this, his assumptions cannot be declared true until they satisfy all the facts which are known to depend on the intimate constitution of fluids. He afterwards finds the equations of equilibrium relative to the surface of separation of two fluids incumbent on one another, and one of the fluids being suppressed, the equation of the free surface of a single fluid. His first memoir closes with this principal conclusion: Capillary phenomena are due to the molecular action resulting from the calorific repulsion, and an attractive force, and modified not only by the form of the surfaces, as in Laplace's theory, but, moreover, by a particular state of compression of the fluid at its superficies. He shows that the variation of density near the surface is extremely rapid, and also that the molecular attractions in fluids as well as in solids extends farther than the calorific repulsion.

152. In the year 1834, Poisson confirmed the above Poisson's conclusion, and the consequences which flow from it with respect to capillary phenomena, in his Nouvelle Theorie de l'Action Capillaire, published in L'Institut for May that year. The object of Poisson in this treatise, is to bring the theory of capillary attraction to the greatest degree of perfection that the power of analysis, and the knowledge of Capillary facts will admit. In the first chapter it is proved that if the fluid within the tube would be horizontal, and there would be neither elevation nor depression. He shows also that we must take account of the variable compression to which the fluid is subject near the surface of the tube, and reaching to the extent of the action due to the solid. From this it would appear that Laplace's theory is somewhat defective, although it explains phenomena. When the fluid is supposed to be incompressible, he finds the equation for the angle of contact to be of the same value as what Dr Young found for it, as also Gauss from the theory of Laplace. If, however, the variation of density be taken into account, the equation will no longer hold.

He next finds the equation of the free surface of a fluid in equilibrium in a capillary space, by taking into account any variation of density that may exist at the fluid surface, although the exact law of variation be unknown; and afterwards he determines on the same principles the equation relative to the contour of the capillary surface.

153. In this theory of Poisson are two principal equations which are the same in form as those of Laplace's theory, and which lead to like results. Generally, however, Poisson has carried his analysis farther than Laplace had done, and so has obtained results more conformable with experiment. Thus, Gay Lussac showed by experiment that the elevation of the lowest point of the capillary surface in a tube 19088 mill. (= 075 in.) diameter, is 15-5861 mill.; on Poisson's theory, it is 15-5829 mill.; and on that of Laplace, 15-5787 mill. In investigating the case where two fluids are superincumbent the one on the other in the same tube, the formulæ of Laplace and Poisson agree, and they are applied to the solution of a curious phenomenon observed by Dr Young. Into a capillary tube containing water, a drop of oil was inserted, when Dr Young saw that the superior surface of the oil depressed itself below the original height of the water. The depression, doubtless, refers to the centre of the capillary surface where it cuts the axis of the tube. If it be assumed that the oil in descending moistened the tube, and that the water did not wet it originally throughout its whole extent, Poisson's theory accounts for the fact.

154. The pressure of fluids modified by capillary action receives considerable attention; he determines both vertical and horizontal pressures on a solid partly immersed in a fluid; and from the calculation of the latter it appears, that when a plate which has its two parallel faces of different substances is the solid immersed, the horizontal pressures on the opposite faces counterbalance each other, and so the solid can have no motion of translation.

155. Various problems which had engaged the attention of previous mathematicians receive more correct solutions by means of Poisson's theory, and are more carefully compared with experiment. The following is one of these results.—When two plates having parallel surfaces are immersed in a fluid, which rises against the surface of one, and is depressed near that of the other, it is found that the fluid surface between them may assume two different forms when the plates are near each other. There is a point of inflection in one which is retained, however near the plates be brought to each other, and in this case they are mutually repulsive, the force being independent of the interval between them; the other is the form noted by Laplace, which contains no inflection, and when it subsists the repulsive changes to an attractive force on making the plates approximate. Poisson explains this by saying, that the first form obtains when the plates, originally at a great distance, are gradually brought near each other; and that the second takes place when, one plate having been previously immersed, the other is inserted into the curved portion of the fluid contiguous to it.

156. Besides the usual problems which this theory solves, Poisson has given the solutions of two which no former mathematician ever attempted; one relates to the form of Poisson fluid poured upon another fluid of greater specific gravity; solves two other has reference to the adhesion of the base of a capillary solid cylinder to a fluid, from which it is raised with its axis vertical. This is perhaps similar to the adhesion of a fluid to a disc, but requiring a different analytical treatment.

157. The last chapter of the treatise contains notes, additions, and a comparison of new experiments. In it are contained the author's views respecting the interior constitution of bodies, particularly fluids, and the nature of molecular forces; as also the treatment of the general equations of the equilibrium of fluids. In this same chapter are several notices of other subjects which are worthy of attention. Thus, the depression of mercury in the barometer cannot be conveniently calculated by the theory unless the ratio of the radius of the tube to one of the constants be either small or great. In other cases the method of Quadratures must be resorted to.

Cashois, professor of physics at Metz, pointed out a method of making barometers with plane or concave surfaces, having observed that, by boiling mercury, the convexity of its capillary surface was diminished, and, by continuing the boiling a sufficient length of time, the mercurial surface in the capillary tube might become concave. Du Long explains this phenomenon. When the mercury is boiling, the surface layer in contact with the air is oxidized, and, mingling with the whole mass, changes its properties so that the action of the particles of mercury on each other, and on those of the tube, or rather on the particles of a thin coating of water which is always interposed between the mercury and the tube, is not the same as before, the change being greater in proportion to the greater quantity of metal oxidized, i.e., in proportion to the boiling (132).

158. Poisson lastly applies his theory to explain the phenomenon of endosmosis. He supposes that the two fluids meet without mixing in the capillary tubes which permeate the membrane, and, by the relation of the molecular forces at their common surface of separation, one prevails over the other, and so passes through to the opposite side of the membrane. But the corresponding phenomenon of exosmosis is unaccounted for (163).

159. With respect to the theories of Laplace and Poisson, Prof. Challis says that M. Poisson's theory will engage his attention of the speculative philosopher, and the simpler mate of the theory of Laplace will be made the vehicle of conveying to Laplace the younger students of science, in an elementary form, the explanation of a numerous and interesting class of phenomena. The same distinguished philosopher makes the following statement in his Report of the British Association for 1836, respecting the views of Laplace and Poisson on capillary phenomena. It does not appear that any exception can be taken to the reasoning in any part of Laplace's theory. The principles may indeed be objected to on the ground that Poisson takes up, viz., that if the molecular constitution of bodies be admitted, there must be a superficial variation of density which that theory takes no account of; as, however, experiment has not yet detected any such variation, and we have no means of assigning the amount of its influence, it would be premature to reject the theory on that ground, especially as the probability is, that the effects which this consideration has on the numerical results of the calculations will at all events be small.

160. In the year 1834 M. Link made several experiments on capillarity. His object was to determine the rarity. Capillary comparative ascents of different fluids by capillary attraction. Instead of using tubes, he employed plates, as in fig. 27, inclined at a small angle with the line of junction vertical, and so arranged the same two plates that they could be dipped and dried, and again placed in the different fluids, and the ascents of the rectangular hyperbola taken. M. Link found that when he used distilled water, nitric acid, a solution of Kali causticum, spirit of wine, sulphuric ether, and rectified sulphuric acid, the ascents were equal between the same two plates. This gentleman in 1836 published in the Annalen experiments which were made on the same subject, but not coinciding with the former, for it was found that different fluids did not ascend to equal heights between the same two plates, and the experiments only partially confirmed the law to which theory leads, of equal ascents of the same fluid between plates of different material well moistened.

161. About the same time Dr Frankenheim of Breslau made experiments on the ascents of fluids in capillary glass tubes, to determine the synaphia, or cohesion of fluid bodies. Let \( h = \) height of ascent, \( r = \) radius of tube, then the specific synaphia varies as \( r(h + \frac{3}{4}r)^2 \).

Illustrations of Capillary Attraction.

162. Many natural phenomena are due to capillary attraction, as, e.g., if the foundation of a building be moist, the moisture will ascend the fine capillary tubes of the stones and mortar, and cause the walls to be damp. So also if a tract of sand or shingle have a hard impermeable base, the rain will, unless drained off, render the district damp and marshy. An excellent illustration of the fact is to be found in the expansion of wood by water. Thus, in the south of France, a large cylinder of freestone, of a proper diameter, and several feet in length, has a number of circular grooves made round its surface; into these grooves wedges of dry wood are driven, and then well soaked with water. After a few hours, the solid cylinder breaks up into rough millstones, which require very little labour to render them fit for the market. Again, it is well known that on a line of rail the rails are secured to the sleepers by means of chairs, and a wooden key binds both rail and chair firmly together. The key must remain immovable in its position; and in order to do this it is first thoroughly steamed, then subjected to a pressure of 12 lbs. on the square inch. The key is now kept in the drying-house till required, when it is easily driven into its cavity, while the moisture, entering its capillary tubes, causes it to expand, and hold rail and chair together with great tenacity. This expansive power of the key has been known to burst asunder its iron chair. Similarly also tight ropes may be rendered still tighter by water being thrown upon them, and consequently may become very serviceable in the arts.

163. But one beautiful application of the principle is to be found in the ascent and descent of sap in trees and plants. M. Dutrochet has named this inward and outward movement endosmose (ἔσως, inwards, and ἐκπορεῖν, impulsion), and exosmose (ἐκ, outwards), respectively. All plants and trees possess cells, which contain liquids of different densities, and which are continually interchanging their contents. When the walls of the cells are thin, the fluid is drunk up very rapidly. The fluids on either side of the membrane differ from each other in density, but have some affinity for the membrane between them as well as for each other. By the endosmotic process, then, a thin liquid passes rapidly upwards in considerable quantity, mixing in its progress with liquids of a denser sort, while, by the exosmotic movement, the latter liquids pass outwards with a slower motion, and in smaller quantity than the former enters. These movements take place in both the living and dead tissue, and they are influenced by the nature of the fluids and of the membrane; they are nicely shown in the case of a unicellular plant. Thus, when one of the cells of the Yeast plant is placed in a dense liquid, it rapidly loses, by an outward movement, its liquid contents, since the cell becomes more or less collapsed; but if in a thin liquid, that liquid will rush up the cell and swell it. It is supposed that the bursting of the seed-vessels of several plants is due to the distention of the cells by endosmose, which causes a curvature in the parts, and ultimately rupture. Endosmose, however, is modified in the living plant by the vital actions going on within the cells, and to these actions are due the continued movements of fluids through the cell walls.

164. The whole process of the endosmotic and exosmotic experiment may be shown by the following experiment: Take a glass vessel filled with alcohol, having its open end two fluids covered tight with a bladder, which has been previously well soaked in water, and is now in close contact with the alcohol. Let this apparatus be immersed in a vessel containing water, the bladder being under the surface. At the end of a few hours, it will be found that so much water has penetrated the bladder-covering and mixed with the alcohol, while a smaller quantity of the latter has made its escape and mingled with the water. On taking up the glass vessel, the surface of the bladder, instead of being flat, will now be puffed outwards and highly convex, so that, if pricked with a needle, a fine column of liquid will be seen ascending to a height of several feet. The ascent of the one is partly caused by capillary attraction, partly also by the chemical affinity of the two fluids, while the greater penetration of the water is owing to the less powerful attraction of the capillary tubes of the bladder on that fluid. The alcohol, being more strongly attracted by the tubes of the bladder, flows out less readily.

The principle may also be illustrated with two different gases; thus, take a common glass tumbler full of air, with the tie carefully over its open end an india-rubber covering, two gases. Place it now under a large bell-glass full of hydrogen, and resting in a dish of water, and it will be found that the hydrogen will gradually penetrate the rubber covering, and mix with the confined air. Such a quantity of hydrogen will mix with the air that the rubber will be much distended, and at last burst. If we reverse the experiment, and put hydrogen in the tumbler, surrounded by air, the rubber will become concave instead of convex.

We may have an idea of the intensity of the forces which take place during these endosmotic and exosmotic movements from the simple fact, that if a glass or porcelain vessel, capable of resisting a pressure of 700 lbs. on the square inch, have its neck well stoppered with a dry piece of wood, and if the projecting end be allowed to dip into water, the liquid will rush up the capillary tubes of the wood, and in a short time break the vessel to pieces. Dutrochet estimated that with a vessel filled with fluid of a density 1-3, and placed in water, the endosmotic force would be equal to a pressure of 4½ atmospheres.

On the Form of Drops.

165. It was observed by M. Monge, that when drops of alcohol fall upon a surface of the same fluid, they do not at first mix with it, but roll over its surface, impinge against drops each other, and are reflected like billiard balls. He observed also an analogous phenomenon in the drops of water which fall from the ears during the rowing of a boat, and during the condensation of the vapour of warm fluids.

In repeating the experiments of Monge, Sir David Brewster found that the phenomena were most beautiful when the capillary tube discharged the drops upon the inclined plane of fluid which is elevated by the attraction of the edge of the cup. They ran down the inclined plane... PART II.—HYDRAULICS.

168. Hydraulics is that branch of the science of Hydrodynamics which relates to fluids in motion. It comprehends the theory of running water, whether issuing from orifices in reservoirs by the pressure of the superincumbent mass, or rising perpendicularly in jets d'eau from the pressure of the atmosphere; whether moving in pipes and canals, or rolling in the beds of rivers. It comprehends also the resistance or the percussion of fluids, and the oscillation of waves.

CHAPTER I.—THEORY OF FLUIDS ISSUING FROM ORIFICES IN RESERVOIRS, EITHER IN A LATERAL OR A VERTICAL DIRECTION.

169. If water issues from an orifice either in the bottom or side of a reservoir, the surface of the fluid in the reservoir is always horizontal till it reaches within a little of the bottom. When a vessel, therefore, is emptying itself, the particles of the fluid descend in vertical lines, as is represented in fig. 43; but when they have reached within three or four inches of the orifice mn, the particles which are not immediately above it change the direction of their motion, and make for the orifice in directions of different degrees of obliquity. The velocities of these particles may be decomposed into two others, one in a horizontal direction, by which they move parallel to the orifice, and the other in a vertical direction, by which they approach that orifice. Now, as the particles about C and D move with greater obliquity than those nearer E, their horizontal velocities must also be greater, and their vertical velocities less. But the particles near E move with so little obliquity that their vertical are much greater than their horizontal velocities, and very little less than their absolute ones. The different particles of the fluid, therefore, will rush through the orifice mn with very different velocities, and in various directions, and will arrive at a certain distance from the orifice in different times. On account of the mutual adhesion of the fluid particles, however, those which have the greatest Motion of velocity drag the rest along with them; and as the former move through the centre of the orifice, the breadth of the issuing column of fluid will be less at op than the width of the orifice mn.

170. That the preceding phenomena really exist when a vessel of water is discharging its contents through an aperture, experience sufficiently testifies. If some small substances specifically heavier than water be thrown into the fluid when the vessel is emptying itself, they will at first descend vertically, and when they come within a few inches of the bottom they will deviate from this direction, and describe oblique curves similar to those in the figure. The contraction of the vein or column of fluid at op is also manifest from observation. It was first discovered by Sir Isaac Newton, and denominated the *vena contracta*.

171. When the orifice is in a thin plate this will be the case, and the contracted part being outside the reservoir may be clearly seen and accurately measured. When the orifice is circular, the vein having attained its least section continues cylindrical in form and has a velocity nearly equal to that due to the charge. Hence the discharge will be the product of the contracted section into the velocity, so that the effect of the contraction is limited to the reduction of the value of the section which enters into the expression of the discharge. The flow will take place as if the real circular opening in the vessel were replaced by one the diameter of which would be equal to the section of contraction, but in which imaginary opening no contraction takes place in the vein issuing out of it. But if to the circular orifice we attach a cylindrical pipe the fluid threads converge on arriving at the junction of the pipe and the wall of the vessel, and so the section of contraction takes place at the entrance of the pipe. But now beyond this section of contraction the sides of the tube attract, and therefore cause the fluid threads to dilate on all sides; and these issuing parallel to each other and to the axis of the pipe, wholly fill the tube, so that the section of the vein equals at its exit the orifice in the vessel's side, but the velocity is not that due to the charge. Were the flow produced solely by the pressure of the fluid, then the velocity at the section of contraction would be that due to the charge, and would diminish in proportion as the fluid vein enlarged itself in virtue of the hydraulic axiom,—that when an incompressible fluid in motion forms a continuous mass, the velocity at all its diverse sections is inversely proportional to the area of the section; the velocity then would cease to diminish when the diverging vein had reached the sides of the tube. Again, when the orifice is at the lowest part of the vertical side of a vessel, the lower edge coinciding with the floor of the vessel, the contraction is then destroyed on that side, and the discharge is somewhat increased. If, instead of a cylindrical adjutage, we make use of a conical mouth-piece, or one converging to a point exterior to the side of the reservoir, then there will be two distinct contractions of the vein, one at the entrance of the adjutage, which will necessarily diminish the velocity due to the charge, and the other at a small distance beyond the issue, the breadth of which will be less than that of the external mouth. Such an adjutage is very regular, throws the water to a greater distance or height, and has a greater flow than that through an orifice in a thin plate. These conical or truncated pyramidal adjutages are frequently used in large manufactories for discharging water upon mill-wheels. The case of the adjutage having its narrow mouth fitted into the side of the reservoir, gives us the largest flow from any kind of tube: it will necessarily have one contracted part at the orifice.

This last kind of adjutage was known to the ancient Romans; several citizens had been granted the privilege of drawing a certain volume of water from the public reservoirs, but as they increased their supply by using this kind of tube, the fraud was detected, and the use of these tubes legally forbidden except at a distance of 52 feet or upwards from the reservoir.

172. Since the direction and velocity of the fluid particles constituting a vein are symmetrical around the parts of the circular orifice, the contracted vein must also be symmetrical in form, and therefore some solid of revolution; it is found to be a conoidal figure. It is actually so, as we shall afterwards see, from observations that have been made upon it. Beyond op (fig. 43), the narrowest section of contraction, this contraction ceases, and the vein continues sensibly cylindrical for some distance till the resistance of the air and the weight of each particle, together with other causes, entirely destroys the form of the curve.

173. The three principal parts of the vein were early investigated, and their ratios determined. Thus the diameters mn and op, and the length ao, are in the ratio of the numbers 1:00 : 0:79 : 0:39, i.e., on this supposition the distance of the contracted vein is about one-half the diameter of the smaller section, and 0:39 of the orifice. But Michelotti, from a mean of more recent experiments, has ranked them in the ratio of 1:00 : 0:787 : 0:498, which D'Aubuisson follows. The ratio, then, of mn to op being as 1:00 : 0:787, the sectional areas of each will be 1: (0:787)^2 = 1:619. The distance also of the contracted vein from the orifice will be a very little more than half the diameter of the orifice, or mo = mn/2 nearly: If s, now, be the contracted section, and S that of the orifice, s = 0:619 S.

174. We proceed now to find the ratio subsisting between the velocities of the particles in the vessel and at the orifice. We may suppose that the mass of fluid is divided into infinitely small horizontal laminae or layers, as NMgh (fig. 43), each particle in the same section having the same velocity, and all descending parallel to each other. Let each particle in the vessel have a velocity v; V, the velocity of escape; A, the horizontal section of the vessel; S or mS, that of the orifice at the bottom, where m is the coefficient of contraction. During a time t, a volume will escape = mS . V . t, but during this same time, the fluid surface of the water will descend through a space Mg, say = velocity of particles into time = v . t, and the corresponding volume will be A . v . t, which must be the same volume as that discharged at the orifice during time t. Therefore A . v . t = mS . V . t; or A . v = mS . V, i.e., v : V = mS : A; or the velocities are inversely as the sections. Hence, if the area mS of the orifice be infinitely small with respect to the area A of the horizontal section of the laminae, the mean velocity of escape will be infinitely greater than that of the laminae; that is, while the velocity at the orifice is finite, that of the laminae will be infinitely small.

175. Before applying these principles to the theory of hydraulics, it may be proper to observe, that several distinguished philosophers have founded the science upon the same general law from which we have deduced the principles of hydrostatics (48). In this way they have represented the motion of fluids in general formulae; but these formulae are so complicated from the very nature of the theory, and the calculations are so intricate, and sometimes impracticable from their length, that they can afford no assistance to the practical engineer.

176. Definition.—If the water issue at mn with the same velocity V, that a heavy body would acquire by falling freely through a given height H, this velocity is said to be due to the height H, and inversely the height H is said to be due to the velocity V. The quantity H is also termed the head or charge under which the flow takes place.

177. Prop. I.—The velocity with which a fluid escapes from a small orifice in the bottom or side of a vessel kept constantly full, equals that attained by a heavy body falling freely from the surface of the fluid to the orifice.

It will be proved in Mechanics, that, when a heavy body is projected obliquely upwards, the composition of the accelerating force of gravity, and the uniform velocity \( V \) of projection, will cause the body, during a time, \( t \), to move in a curve called a parabola, the equation to which is found to be \( y^2 = 4kx \), where \( k \) = height due to velocity, \( x \) the absciss in the direction of gravity, \( y \) the ordinate parallel to the direction of projection, or \( x \) and \( y \) are the co-ordinates for any point of the curve.

Now, this is true, whatever be the nature of the body; and, therefore, also for a jet of water issuing from an orifice. Let the orifice be opened in the side at \( A \) (fig. 44), then

\[ y^2 = 4kx \]

gives us \( h = \frac{y^2}{4x} \) = height due to velocity of exit. This value of \( h \) may be easily calculated by placing a vertical graduated stick alongside the vessel, and having its zero point at the centre of the orifice, with another moving at right angles, and also graduated; this last will give us any ordinate, as \( PM \) or \( CD \); and therefore, also, the corresponding abscissa \( AP \) or \( AC \). Therefore, by measuring the ordinate and its corresponding absciss, the height \( h \) due to the velocity of exit may be determined from the equation to the parabola. But the velocity which a body (by mechanics) falling freely attains, after passing through a certain height \( h \), is represented by \( V = \sqrt{2gh} \). Therefore, the velocity of issue at the orifice is that due to the height \( BA \).

If the orifice be at the bottom, the velocity will be that due to a fall from the surface to the orifice.

178. Con. 1.—As fluids press equally in all directions, the preceding proposition will hold true when the orifices are at the sides of vessels, and when they are formed to throw the fluid upwards, either in a vertical or an inclined direction, provided that the orifices are in these several cases at an equal distance from the upper surface of the fluid.

Con. 2.—When the fluid issues vertically, it will rise to a height equal to the perpendicular distance of the orifice from the surface of the fluid; owing to the resistance of the air, however, and the friction of the issuing fluid upon the sides of the orifice, jets of water do not exactly rise to this height.

179. If \( H \) be the height from which a falling body is dropped, then the velocity which it acquires after a certain time, will be equal to that of any particle of the fluid as it issues from an orifice, or \( V = \sqrt{2gH} \), where \( g \) = the velocity, which a body falling freely would attain at the end of a unit of time. If the mean velocity of all the particles be that due to a charge \( H \), then the theoretic velocity is \( V = \sqrt{2gH} \).

Supposing, then, that the fluid threads issue from the orifice parallel to each other, and knowing the velocity of the charge of one particle, the theoretic discharge, or that volume which escapes in a unit of time, will evidently be the volume of a prism, having for its base that of the orifice, and that velocity for height. Calling then \( S \) the area of the orifice, \( S \cdot V \) represents the theoretic discharge, or \( S \cdot V = S \sqrt{2gH} \).

But the actual discharge is always less than that theoretically determined; and in order to have an exact idea of this, take a perpendicular section of the fluid vein at a small distance from the orifice; then since the discharge is manifestly equal to the sectional area into the mean velocity of any thread at the instant it is crossing that section, it is clear that if this section were equal to that of the orifice, and if the velocity were that due to the charge, then the actual would be equal to the theoretic discharge. But, whether from the section of the vein being less than that of the orifice, as in the case of a flow through a thin plate, or from the velocity being less than that due to the charge, as in cylindrical pipes; or from a diminution in both section and velocity, as in the case of conical adjutages, the result always is, that the actual is less than the theoretic discharge, and so the latter must always be multiplied by some fractional number, so as to obtain the former. Let this fractional number be \( m \), the coefficient of contraction, then the actual volume discharged \( = mS \sqrt{2gH} \).

180. The reader will probably be surprised when he finds in some of our elementary works on hydrostatics, that the velocity of the water at the orifice is only equal to that which a heavy body would acquire by falling through half the height of the fluid above the orifice. This was first maintained by Sir Isaac Newton, who found that the diameter of the rena contracta was to that of the orifice as 21 to 25. The area therefore of the one was to the area of the other as \( 21^2 \) to \( 25^2 \), which is nearly the ratio of 1 to \( \sqrt{2} \). But by measuring the quantity of water discharged in a given time, and also the area of the rena contracta, Sir Isaac found that the velocity at the rena contracta was that which was due to the whole altitude of the fluid above the orifice. He therefore concluded, that since the velocity at the orifice was to that at the rena contracta as 1 : \( \sqrt{2} \), and the latter velocity was that which was due to the whole altitude of the fluid, the former velocity, or that at the orifice, must be that which is due to only half that altitude, the velocities being as the square roots of the heights. Now the difference between this theory and that contained in the preceding proposition may be thus explained. The velocity found by the preceding proposition is evidently the vertical velocity of the filaments at any point, which being immediately above the centre of the aperture are not diverted from their course, and have therefore their vertical equal to their absolute velocity. But the vertical velocity of the particles below and above the former is much less than their absolute velocity, on account of the obliquity of their motion, and also on account of their friction on the sides of the orifice. The mean vertical velocity, consequently, of the issuing fluid will be much less than the vertical velocity of the first particle, that is, than the velocity found by the above proposition, or that due to the full height. Now the velocity found by Sir Isaac Newton from measuring the quantity of water discharged, was evidently the mean velocity, which ought to be less than the velocity given by the preceding proposition; the two velocities being as 1 : \( \sqrt{2} \), or as 1 : 1.414. The theorem of Newton therefore may be considered as giving the mean velocity at the orifice, while the proposition gives the velocity of the particles at the surface, or the velocity at the rena contracta.

181. Prop. II.—To find the volume discharged from a very small orifice in the side or bottom of a reservoir kept constantly full, the time of discharge and the altitude of the fluid being given.

Let \( S \) = area of orifice, \( Q \) = volume discharged in time \( T \), and call \( H \) the height of the fluid, or the constant charge due to the constant height. Then, since the actual discharge in a unit of time or one second, is \( Q = mS \sqrt{2gH} \). Motion of (179), therefore, the volume discharged during T units Fluids, &c., or seconds, will be QT, or \( Q = m ST \sqrt{2gh} \).

Therefore, \( Q, m, S, T \) or \( H \), may be easily determined, all the others being given.

The diminution of discharge conceived as arising from the sectional area or velocity diminishing, is always a consequence of the contraction of the vein in passing through the orifice. It is the variation in the actual, from the theoretical discharge, that gives rise to \( m \), the coefficient of contraction, equal to the ratio of the actual to the theoretical discharge. It is a very important number, for, on its exactness depends the accuracy of formula when practically applied to the flow of water; \( m \) is always determined experimentally.

182. It is supposed in the preceding proposition that the orifice in the side of the vessel is so small, that every part of it is equally distant from the surface of the fluid. But when the orifice is large, like M (fig. 45), the depths of different parts of the orifice below the surface of the fluid are very different, and consequently the preceding formula will not give very accurate results. If we suppose the orifice M divided into a number of smaller orifices, \( a, b, c \), it is evident that the water will issue at \( a \) with a velocity due to the height Da, the water at \( b \) with a velocity due to the height Eb, and the water at \( c \) with a velocity due to the height Ec. When the whole orifice, therefore, is opened, the fluid will issue with different velocities at different parts of its section. Consequently, in order to find new formulae expressing the quantity of water discharged, we must conceive the orifice to be divided into an infinite number of areas or portions by horizontal planes; and by considering each area as an orifice, and finding the quantity which it will discharge in a given time, the sum of all these quantities will be the quantity discharged by the whole orifice M.

183. Prop. III.—To find the volume discharged by a rectangular slit in the vertical side of a vessel, kept constantly full.

Let M (fig. 46) be the vessel, of a vertical height AB, equal to the depth, and \( l \) the width of the slit. If now we have a series of elementary rectangular orifices placed vertically in the side of M, then if the lowest be at a depth AB or \( H \), below the fluid surface, the velocity with which the fluid issues from this orifice will be \( \sqrt{2gh} \). Now we may make BG equal to this quantity, and this ordinate will represent the velocity, while the absciss does the depth; so also we may take any other point C, at a depth C or AC = \( x \), and the velocity of issue will be represented by CD = \( y = \sqrt{2gx} \); the same will be the case for all other points in AB, and, evidently, if we join all the ordinate points as C, D, &c., we shall have a parabolic curve; and as \( y^2 = 2gx \), where \( 2g \) or 64' feet is the parameter, we shall have the velocity of a fluid thread issuing from a reservoir at any point equal to the ordinate of a parabola, the parameter of which is twice the measure of the accelerating force of gravity, with a depth under the surface as the abscissa.

Suppose now that all the rectangular orifices in the depth AB were opened, making a rectangular slit ABPR, with a width of \( l \) units, the discharge through this opening may thus be obtained. Since the surface of the jet has been proved to be parabolic, we shall evidently have a volume of water which shall be the frustum of a paraboloid, the side of which will be ABGDA, and width \( l \); and since the area of a parabola = \( \frac{2}{3} \)ds of the rectangle contained by AB and BG, and \( AB \cdot BG = H \cdot \sqrt{2gh} \), therefore the volume, or the discharge for the rectangular opening, is \( \frac{2}{3} l \cdot H \cdot \sqrt{2gh} = \frac{2}{3} l \cdot H \cdot \sqrt{H} \cdot \sqrt{2g} \).

184. Prop. IV.—To find the volume discharged from a rectangular orifice.

This volume may be deduced from the last. Let the rectangular orifice have a width equal to \( l \), and a depth \( = OB \); call \( AO = h \), therefore, the volume of discharge by the frustal paraboloid AOUA = \( \frac{2}{3} l \cdot h \cdot \sqrt{2gh} \), or \( = \frac{2}{3} l \cdot h \cdot \sqrt{h} \cdot \sqrt{2g} \). Hence the discharge by the rectangular orifice OBPN = the discharge by ABPR, less that by AONR = \( \frac{2}{3} l \cdot H \cdot \sqrt{H} \cdot \sqrt{2g} - \frac{2}{3} l \cdot h \cdot \sqrt{h} \cdot \sqrt{2g} = \frac{2}{3} l \cdot \sqrt{2g} (H \cdot \sqrt{H} - h \cdot \sqrt{h}) \).

185. Prop. V.—To determine the mean velocity from the above slit (Prop. III.)

The mean velocity of the fluid from the rectangular opening ABPR, may be thus obtained. Let K be the point from which would issue a fluid thread with the mean velocity; and making the depth AK = \( z \), the mean velocity for this thread will be \( \sqrt{2gz} \), which, multiplied by the area or \( l \cdot H \), will give the volume discharged with the mean velocity from the opening ABPR, \( = l \cdot H \cdot \sqrt{2g} \cdot \sqrt{z} \). But we have already seen, that the total discharge from the opening itself \( = \frac{2}{3} l \cdot H \cdot \sqrt{H} \cdot \sqrt{2g} \);

\[ \therefore \quad l \cdot H \cdot \sqrt{2g} \cdot \sqrt{z} = \frac{2}{3} l \cdot H \cdot \sqrt{H} \cdot \sqrt{2g}; \]

or, \( \sqrt{z} = \frac{2}{3} \sqrt{H} \), or \( z = \frac{2}{3} H \). Consequently, since the mean velocity, which call \( V = \sqrt{2gz} \);

\[ \therefore \quad V = \sqrt{2g} \cdot \frac{2}{3} \cdot H = \frac{2}{3} \sqrt{2g} H, \]

that is to say, that the mean velocity is \( \frac{2}{3} \)ds that with which the lowermost fillet issues; and so KW, which represents the former velocity, will be \( \frac{2}{3} \)ds of BG, which represents the latter.

186. Prop. VI.—To determine the mean velocity of the rectangular orifice OBPN of last figure.

The mean velocity of the fluid from the rectangular orifice OBPN, the depth being \( (H - h) \), and width \( l \), may also be found. Call \( z' \) the depth of that point from which would issue a fluid thread with the mean velocity; then the discharge from OBPN with the mean velocity \( = (H - h) \cdot l \cdot \sqrt{2gz'} \); but the total velocity from this orifice is \( \frac{2}{3} l \cdot \sqrt{2g} (H \cdot \sqrt{H} - h \cdot \sqrt{h}) \);

\[ \therefore \quad (H - h) \cdot l \cdot \sqrt{2gz'} = \frac{2}{3} l \cdot \sqrt{2g} (H \cdot \sqrt{H} - h \cdot \sqrt{h}); \]

or, \( (H - h) \cdot \sqrt{z'} = \frac{2}{3} (H \cdot \sqrt{H} - h \cdot \sqrt{h}) \);

\[ \therefore \quad z' = \frac{2}{3} \left( \frac{H \cdot \sqrt{H} - h \cdot \sqrt{h}}{H - h} \right)^2. \]

187. We have hitherto supposed that the fluid surface has always been maintained at the same level, but suppose now that the vessel receives no supply, or receives less than it discharges by an orifice in the bottom, it is clear that the fluid will gradually near the bottom of the tank and empty itself. The laws of discharge then will be somewhat different from those previously considered. The vessels con- 188. Prop. VII.—Find the height due to the velocity of the fluid at its point of discharge, when the fluid surface is not maintained always at the same level.

Here \( V \) is dependent on a constantly varying height, and on \( v \), the velocity acquired in the descent of the layers. Let then \( H_1 \) be the generating height due to the velocity \( V \) of the fluid at the point of discharge, and \( H \) always the actual height in the vessel; hence,

\[ H = H + \frac{v^2}{2g} = H + \frac{m^2 S^2}{2g A^2} \quad \text{(by 174)} = H + H_1 \frac{m^2 S^2}{A^2}; \]

\[ \therefore \quad H = H + H_1 \frac{m^2 S^2}{A^2}, \quad \text{or} \quad H_1 \left(1 - \frac{m^2 S^2}{A^2}\right) = H, \quad \text{and} \quad H_1 \text{ is known}. \]

\( m, S \) is generally small when compared with \( A \), so also much smaller will \( m^2 S^2 \) be in comparison with \( A^2 \), and therefore \( \frac{m^2 S^2}{A^2} \) may be neglected; hence, \( H_1 = H \), i.e., the velocity of escape at any given instant is that due to the actual height of water in the vessel at that moment; a result similar to that arrived at when the fluid surface was maintained at a constant level. But the above is true only in the case of those layers or sections which are supposed to remain parallel during their descent. As soon as the parallel laminae come within the sphere of action of the orifice, the nature of the motion is very complicated, and indeed unknown.

189. Prop. VIII.—To determine the nature of the motion which goes on within a vessel emptying itself.

Let \( AD \) (fig. 47) be a vessel filled with fluid to the brim \( AB \), having an orifice \( D \) in the side \( BD \); suppose \( BD \) to be divided into equal parts, \( Ba, ab, bc, \ldots \); also let \( P \) be taken opposite the centre of the orifice, and take a vertical line, \( PH = BD \), dividing it into the same number of equal parts as \( BD \). Now, that a body be projected from \( P \) vertically so as to reach \( H \), it must have the same velocity impressed upon it as the body would acquire on arriving at \( P \), when dropped freely from \( H \), for the velocity which the body has at \( P \) in ascending, is the same as it has when descending from \( H \). Hence, on arriving at the points \( a, b, c, \ldots \), the velocities will be respectively \( \sqrt{2gH_a}, \sqrt{2gH_b}, \sqrt{2gH_c}, \ldots \); or as \( \sqrt{Ha}, \sqrt{Hb}, \sqrt{Hc}, \ldots \). Therefore, also, as the surface \( AB \) passes the points \( a, b, c, \ldots \), the velocities of escape at the orifice will be proportional to \( \sqrt{Da}, \sqrt{Db}, \sqrt{Dc}, \ldots \); that is, as the corresponding equal velocities \( \sqrt{Ha}, \sqrt{Hb}, \sqrt{Hc}, \ldots \). Hence, as the vessel empties itself, the velocity of discharge gradually decreases from a maximum, \( \sqrt{2gDB} \), to a minimum as zero, following the same law as does a body when thrown vertically upwards. Each is an example of uniformly retarded motion: the velocity of escape is governed by the same law.

190. Cor. 1.—As the velocities of falling bodies are as the square roots of the heights through which they fall (see Mechanics), the velocities of fluids issuing from a very small orifice are as the square roots of the altitude of the water above these orifices. As the quantities of fluids discharged are as the velocities, they will also be as the square roots of the altitude of the fluid. This corollary holds true of fluids of different specific gravities, although Be-

lidor (Architect. Hydraul., tom. i., p. 187) has maintained Motion of the contrary; for though a column of mercury presses with Fluids, &c., fourteen times the force of a similar column of water, yet the column of mercury which is pushed out is also fourteen times as heavy as a similar column of water; and as the resistance bears the same proportion to the moving force, the velocities must be equal.

191. Cor. 2.—When a vessel is emptying itself, if the area of the laminae into which we may suppose it divided, be everywhere the same, the velocity with which the surface of the fluid descends, and also the velocity of efflux, will be uniformly retarded. For as the velocity \( V \) with which the surface descends is to the velocity \( v \) at the orifice, as the area \( a \) of the orifice to the area \( A \) of the surface, then \( V : v = a : A \); but the ratio of \( a : A \) is constant, therefore \( V \) varies as \( v \), that is, \( V : V' = r : r' \); but, \( v : v' = \sqrt{h : \sqrt{H}}, h \) being the height of the surface above the orifice, therefore \( V : V' = \sqrt{h : \sqrt{H}} \). But this is the property of a body projected vertically from the earth's surface, and as the retarding force is uniform in the one case (see Mechanics), it must also be uniform in the other.

192. With respect to the volume discharged, we know from mechanics that when a body starts with a certain velocity which is gradually lost altogether, it describes one-half the space only that it would have described in the same time had it continued to move uniformly with the velocity with which it began. Similarly, also, the volume of fluid in the vessel may be looked upon as a cylindrical or prismatic mass, having its base equal to the dimensions of the orifice, and its height, the space which the first issuing particle would describe with a motion uniformly retarded, and identical with that by which the discharge takes place; if, now, the particle had retained its first velocity it would have doubled its space, or the height of the cylindrical or prismatic mass would have been doubled, and therefore also the volume of discharge from the orifice would have been doubled. Hence, then, it appears that the volume of fluid passing through an orifice at the bottom of a prismatic or cylindrical vessel, the surface of which is gradually nearing the orifice till the vessel empties itself, having received no supply, is equal to one-half that which would be given during the time of complete discharge, if the flow had taken place under a constant charge equal to the primary.

193. Cor. 3.—If a cylindrical vessel be kept constantly full, twice the quantity contained in the vessel will run out during the time in which the vessel would have emptied itself. For the space through which the surface of the fluid at \( L \) would descend if its velocity continued uniform, being \( 2LM \), double of \( LM \), the space which it actually describes in the time it empties itself, the quantity discharged in the former case will also be double the quantity discharged in the latter; because the quantity discharged when the vessel is kept full, may be measured by what the descent of the surface would be, if it could descend with its first velocity,—\( M \) being regarded as the only orifice in and at the bottom of the vessel (fig. 47).

194. Prop. IX.—To find the time required to empty a vessel.

Suppose that \( H \) is its charge, \( A \) the horizontal section of the vessel, and \( T \) the time of complete discharge. The volume of water contained in the vessel above the orifice, and therefore the whole quantity discharged during the time, will be \( A.H \). But (by 192) the volume which would have been discharged in the same time under the constant charge \( H \) would have been \( 2A.H \). This same volume of discharge during time \( T \) equals \( mST \sqrt{2gH} \), where again \( S \) or \( mS \) is the area of the orifice; whence

\[ 2A.H = mST \sqrt{2gH}; \quad \text{or}, \] 195. Cor.—If \( T \) be the time that the volume \( A.H \) would take to flow out, when the head \( H \) was constant, then (by 181),

\[ A.H = mS\sqrt{2gH}; \quad \therefore \quad T = \frac{A.H}{mS\sqrt{2g}} = \frac{A}{mS} \sqrt{\frac{H}{2g}}; \]

\(\because\) from the above \( T = 2T' \); that is, the time which a prismatic vessel takes to be completely discharged, is double that in which the same volume would flow out, if the head had remained constantly the same as it was at the commencement of the discharge.

196. Prop. X.—To find the time that the fluid surface takes to pass through a given depth.

Let \( t \) be this time for the level to descend a vertical depth \( a \). Now we have already seen that the time during which the whole volume will discharge itself is

\[ \frac{2A}{mS} \sqrt{\frac{H}{2g}}, \]

the charge at the beginning of the flow being \( H \); and by putting \( H - a = h \) for the charge or head at the end of the time \( t \), the time during which the volume \( hA \) would be entirely discharged is

\[ \frac{2A}{mS} \sqrt{\frac{h}{2g}}. \]

Now, we have the time in which the whole volume with a head \( H \), and that in which a head equal to \( h \) or \( (H - a) \), will be discharged; therefore the time in which a head equal to \( a \) will be discharged, is the difference between these two times; or,

\[ t = \frac{2A}{mS} \sqrt{\frac{H}{2g}} - \frac{2A}{mS} \sqrt{\frac{h}{2g}} = \frac{2A}{mS} \sqrt{\frac{H-h}{2g}}. \]

197. Cor.—The quantity of fluid discharged in the given time \( t \) may be found by measuring the contents of the vessel between the planes, the descent of the surface, or the depth \( a \), being known.

198. Prop. XI.—To find the volume discharged in a given time, and the depth that the level reaches after a certain time.

From the last expression for the time required to empty a vessel, we have

\[ \frac{tmS\sqrt{2g}}{2A} = \sqrt{H - \sqrt{h}}; \quad \text{or,} \]

\[ \left( \frac{tmS\sqrt{2g}}{2A} \right)^2 + 2 \left( \frac{tmS\sqrt{2g}}{2A} \right) \sqrt{h} + h = H; \]

\[ \therefore \quad tmS\sqrt{2g} \left( \frac{tmS\sqrt{2g}}{4A} + \sqrt{h} \right) = H - h = a = \text{the depth that the level reaches in a certain time.} \]

If now we multiply this depth by \( A \), the horizontal section of the vessel, we shall have the volume discharged in the time \( t \); \(\therefore\) volume discharged, say,

\[ Q = (H - h)A = tmS\sqrt{2g} \left( \frac{tmS\sqrt{2g}}{4A} + \sqrt{h} \right). \]

199. Prop. XII.—To find the mean hydraulic charge.

The prismatic vessel fig. 48 receives no supply, but discharges water through an orifice, the area of which is \( S \); the water at the commencement of the flow has a head \( H \), and at the end of the time \( T \) has a head equal to \( h \); the mean hydraulic charge \( H' \) is required, by which, ceteris paribus, the same volume of water would have been discharged.

The volume with this mean head \( H' \) will, in time \( T \), be

\[ Q = mS\sqrt{2g} \sqrt{H'} = (H - h)A; \quad \text{and the time in which this took place was } T = \frac{2A}{mS\sqrt{2g}} (\sqrt{H} - \sqrt{h}). \]

Substitute this value of \( T \) in the above expression for the volume, then,

\[ Q = mS\sqrt{2g} \cdot \frac{2A}{mS\sqrt{2g}} (\sqrt{H} - \sqrt{h}) \cdot \sqrt{H'} = (H - h)A; \]

\[ \therefore \quad \sqrt{H'} = \frac{H - h}{2\sqrt{H} - \sqrt{h}} = \frac{\sqrt{H} + \sqrt{h}}{2}. \]

200. Cor.—If \( h = 0 \), \( H' = \frac{H}{4} \).

Dr Young made use of this proposition in determining by experiment the coefficient of contraction at an orifice. See his experiments (383).

201. Prop. XIII.—If a prismatic vessel receive a constant supply, less however than that discharged, to find the time that the surface will take to fall through a given height.

Let the letters stand for the same things as before, and further, let the volume of water entering the basin in one second be \( q \), and \( x \) the height through which the water descends in a given time \( t \), then \( dx \) will be the height in the infinitely small time \( dt \), and \( Adx \) would represent the volume during the time \( dt \) if the vessel received no supply; but since in one second it receives a supply \( q \), it will in time \( dt \) receive \( qdt \) of a supply. Therefore the actual discharge during time \( dt = Adx + qdt \); but this same volume (by 181) is expressed by \( mS\sqrt{2g}(H-x) \); hence

\[ Adx + qdt = mS\sqrt{2g}(H-x); \quad \text{let } H - x = h, \]

then \(-dx = dh\), since \( H \) is constant. Therefore,

\[ qdt - Adh = mS\sqrt{2g}dh; \quad \therefore \quad dt = \frac{-Adh}{mS\sqrt{2g} - q} = \frac{-Adh}{mS\sqrt{2g} - q}, \]

where \( mS\sqrt{2g} - q = y \): by differentiating, we have,

\[ \frac{mS\sqrt{2g}}{2\sqrt{h}} = \frac{dy}{dh}; \quad \text{or, } \frac{dh}{dy} = \frac{mS\sqrt{2g}}{2\sqrt{h}} = \frac{mS\sqrt{2g}}{2\sqrt{h}} = \frac{mS\sqrt{2g}}{2\sqrt{h}}; \]

or, \( dh = \frac{dy}{mS\sqrt{2g}} \); \(\therefore \quad dt = \frac{-Ady}{mS\sqrt{2g}} = \frac{-A}{mS\sqrt{2g}} (dy + qy)\).

Therefore,

\[ \int dt = \frac{-A}{mS\sqrt{2g}} \int (dy + qy); \quad \text{or, } t = \frac{-A}{mS\sqrt{2g}} (y + q \text{ hyp. log. } y) + C. \]

Since when \( t = 0 \), \( x = 0 \), and so \( H \) will be \( h \);

\[ \therefore \quad C = \frac{-A}{mS\sqrt{2g}} (y + q \text{ hyp. log. } y), \quad \text{and by substitution} \]

\[ = \frac{-2A}{(mS\sqrt{2g})^2} \left[ mS\sqrt{2g} \sqrt{H} - q + q \text{ hyp. log. } (mS\sqrt{2g} \sqrt{H} - q) \right]; \]

Therefore \( t = \)

\[ \frac{-2A}{(mS\sqrt{2g})^2} \left[ mS\sqrt{2g} \sqrt{H} - q + q \text{ hyp. log. } (mS\sqrt{2g} \sqrt{H} - q) \right] \]

\[ + \frac{-2A}{(mS\sqrt{2g})^2} \left[ mS\sqrt{2g} \sqrt{H} - q + q \text{ hyp. log. } (mS\sqrt{2g} \sqrt{H} - q) \right] \]

\[ = \frac{-2A}{(mS\sqrt{2g})^2} \left[ mS\sqrt{2g} \sqrt{H} - q + q \text{ hyp. log. } (mS\sqrt{2g} \sqrt{H} - q) \right]. \]

Make \( q = 0 \), then the expression becomes identical with 196.

202. If the converse question were required, viz., given the time, to find the height to which the level of the water would descend; we should be doing the same thing as if we were to determine the charge \( h \) at the end of the given time, and deduct it from \( H \), the head at the beginning of the charge. In order to find \( h \), successive values of it, i.e., \( H - x \), must be substituted in the above equation, and by the process of trial and error, that value which satisfies the equation will be the one required.

203. Prop. XIV.—To find the time of discharge from a vessel other than prismatic.

The determination of the time in this case is much more complicated than for any other form of vessel, and it is sometimes impossible. The fundamental equation, however, for this purpose is

\[ A \cdot dx = mSdt\sqrt{2g(H-x)}; \quad \therefore dt = \frac{A \cdot dx}{mS\sqrt{2g(H-x)}}. \]

The form of the vessel, or a horizontal section of its area, is variable. In order, therefore, that the above value for \( dt \) may be integrable, \( x \) must be known in terms of \( x \), which can only be the case when the interior curvature of the basin is given. If this cannot be ascertained, we must proceed by approximations and by parts. For this purpose, the basin must be divided into horizontal sections of small depth, each of which is to be regarded as a prism, and the time determined that any one of these takes to pass out, making use of the above-mentioned formulae. The sum of all these partial times will give the required time that the surface takes to descend a height equal to the sum of the heights of the prisms.

204. Prop. XV.—Water is discharged from one reservoir into another, find the volume discharged and the time required to fill up to different levels in one of the reservoirs.

Three cases will require to be considered.

First Case.—Suppose the orifice covered with water on both sides or faces, the levels in the reservoirs M and N (fig. 49) remaining constant; then, since the charge is \( H - h \), where \( H \) is the charge for M, and \( h \) that for N, the volume discharged will be the same as if it had issued out into the open air with a velocity equal to the difference of the charges on each face. Hence, volume discharged in one unit, as a second, is \( Q = mS\sqrt{2g(H-h)} \), where again S or mS is the area of the orifice.

Second Case.—Let the level in M, the upper basin, be constant, while the lower basin N of a given area receives it without any loss, to determine the time that the level of N will reach the level of the upper basin, or a given height. This problem is the converse of 196, where the discharge took place in air, and the surface of the water above the orifice descended with a uniformly retarded motion. Here, however, the surface of the basin is urged from below upwards by a force equal to the difference between the levels of the two basins, decreasing in the same proportion as the charge decreased in the former case, but at the same time, rises with a motion uniformly retarded, and it will require the same time to traverse the same space, under these similar pressures.

Suppose, then, that \( H \) is the pressure or charge AC, at the commencement, and \( h \) the charge at the end of the time \( t \); let also A be the horizontal section of the vessel being filled up, and m and S as before. In order, therefore, that the water may fill N up to DE, \( t = \frac{2A}{mS\sqrt{2g}}(\sqrt{H} - \sqrt{h}) \); and for the time up to AF, the level of M, we have \( T = \frac{2A}{mS\sqrt{2g}}\sqrt{H} \). These formulae are important, more especially in determining the time in which canal locks, &c., may be filled; and also for assigning the area of sluice-way required to fill in a given time.

Third Case.—Suppose that the level of the water is variable in each vessel. This case is that of two reservoirs of different levels, in communication with each other, each limited in area and receiving no supply, so that as one surface falls the other rises.

Let K and L (fig. 50) be two such tanks, communicating with each other by the wide pipe EF, a cock or sluice-door being at G. Let the cock G be unopened, then the level of K is AB, and that of L is CO. Suppose now, that G is open, then, at the end of a certain time, AB has fallen to MN, while the corresponding rise in L is PQ. We want, then, to determine the relation between the two heights at a given time, and conversely, from the given difference of height in the two vessels, to deduce the time corresponding to a given discharge.

Let \( t \) be the time, \( BE = H \), \( CF = h \), \( NE = x \), \( PF = y \), while the horizontal sections of K, L, and the communicating pipe, are respectively denoted by A, B, and s, and the coefficient of contraction, together with the resistance of the water passing through the pipe EF, are represented by \( m \).

Now, while during the very small time \( dt \), the level of K has lowered by the quantity \( dx \), and that of L has risen by \( dy \); and since as \( x \) diminishes while \( y \) and \( t \) increase, we have \( Adx = -Bdy \); \( \therefore Adx = -ms\sqrt{2g(x-y)} \) (by 181); the one being positive while the other is negative; hence \( dt = -\frac{Adx}{mS\sqrt{2g}(x-y)} \), hence also \( \int Adx = -\int Bdy \); or \( Ax + By = AH + Bh \), since in the limit \( x = H \), and \( y = h \); \( \therefore y = \frac{A(H-x)}{B} + h \). Let this value for \( y \) be substituted in the value for \( dt \), and we have \( dt = \frac{-Adx}{mS\sqrt{2g}(x-y)} \); hence \( \int Adx = \frac{A\sqrt{B}ds}{mS\sqrt{2g}(x-y)} \); \( \therefore t = \frac{2A\sqrt{B}}{mS\sqrt{2g}(A+B)} \left( \sqrt{B(H-h)} - \sqrt{(A+B)x-AH-Bh} \right) \).

By integration; since \( x = H \) in the limit, and \( t = 0 \).

Let the time required be now that in which the levels of K and L form a straight line, then \( x = y \);

\( \therefore (A+B)y = (A+B)y = AH + Bh \); or, \( x = y = \frac{AH + Bh}{A + B} \);

and if this value of \( x \) be substituted in the value of \( t \), we have \( t = \frac{2AB\sqrt{H-h}}{mS\sqrt{2g}(A+B)} \). Hence, for the same value of \( (H-h) \), \( t \) is the same whether A be the horizontal section of the basin that lowers, and B that of the other, the surface of which rises, or vice versa, B that which falls, and A that which rises.

205. Prop. XVI.—To find the velocity and volume discharged, of water issuing from an orifice in the side of a vessel, when the fluid has an antecedent velocity.

When water is continuously poured into a vessel containing fluid, or if the water in the vessel be moving in the direction of the orifice, then, the particles will approach to and issue from the orifice in virtue of the pressure of the fluid mass above it, together with the additional velocity that they had while they entered into the sphere of action of the orifice. Therefore, the additional velocity will be that arising from the height of a column, which would exert the same effect as that which the fluid previously had. Let this velocity be equal to \( u \), which is \( \frac{u^2}{2g} \), or that due to the height of a column \( h \); call the depth of the fluid \( h \), then the velocity of efflux for one particle or thread will be that due to the vertical height \( h + h' \). Hence, velocity of escape is

\[ V = \sqrt{2g(h + h')} = \sqrt{2g\left(h + \frac{u^2}{2g}\right)} = \sqrt{2gh + u^2}. \]

Call \( S \), or \( mS \), the area of orifice, then volume discharged in a unit of time is

\[ Q = mS\sqrt{2gh + u^2} = mS\sqrt{2g\left(h + \frac{u^2}{2g}\right)}. \]

the volume discharged in \( T \) units of time will be,

\[ Q \cdot F, \text{ or } Q = mST \cdot \sqrt{2g\left(h + \frac{u^2}{2g}\right)}. \]

206. Cor.—Take \( h = 34 \) feet, or a water column which would balance the atmospheric column, then \( V = \sqrt{2g(h + 34)} \), which expresses the velocity with which water is projected into a vacuum, the column being \( h \).

207. Prop. XVII.—If two cylindrical vessels be filled with water, the times in which the fluid surfaces will fall through given heights will be directly as the compound ratio of their bases, and the difference between the square roots of the altitudes of each surface at the beginning and end of its motion, and inversely as the areas of the orifices.

Let \( A \) and \( A' \) be the areas of the cylindrical vessels, \( a \) and \( a' \) the spaces passed through by the fluid surfaces in the corresponding times \( T \) and \( T' \), \( a \) being \( H - h \); \( a' = H' - h' \); where \( H, H', h, h' \) are the heights of the fluid surfaces respectively at the beginning and end of the times; \( S \) and \( S' \) or \( mS \) and \( mS' \) the areas of the orifices of \( A \) and \( A' \), the coefficients being regarded as equal: then, by Prop. X, 196,

\[ T = \frac{2A}{mS} \cdot \frac{1}{\sqrt{2g}} (\sqrt{H} - \sqrt{h}), \quad \text{and} \quad T' = \frac{2A'}{mS'} \cdot \frac{1}{\sqrt{2g}} (\sqrt{H'} - \sqrt{h'}). \]

\[ \therefore T : T' = \frac{2A}{mS} \cdot \frac{1}{\sqrt{2g}} (\sqrt{H} - \sqrt{h}) : \frac{2A'}{mS'} \cdot \frac{1}{\sqrt{2g}} (\sqrt{H'} - \sqrt{h'}) = \frac{A}{S} (\sqrt{H} - \sqrt{h}) : \frac{A'}{S'} (\sqrt{H'} - \sqrt{h'}). \]

208. Cor.—Hence the time in which two cylindrical vessels full of water will empty themselves, will be directly as the compound ratio of their bases, and the square roots of their altitudes, and inversely as the areas of their orifices.

For, in the last article, \( \sqrt{h} = \sqrt{H} = 0 \), therefore,

\[ T : T' = \frac{A}{S} \sqrt{H} : \frac{A'}{S'} \sqrt{H'}, \]

which is the same ratio as would have been obtained had we used the formula of Prop. IX, 194.

209. Prop. XVIII.—To find the horizontal distance to which water will spout from an orifice made in the side of a vessel, when projected horizontally.

Let AC be the vessel full of fluid; suppose that N is an orifice in its side, at a depth BN below AB; NT the path of the jet. Let also \( v \) be the velocity with which water issues from N, and \( t \) the time of its falling down the perpendicular distance NC. Then

\[ v = \sqrt{2g \cdot BN} \quad \text{(Prop. I, 177)}, \quad \text{and} \quad t = \frac{\sqrt{2g \cdot NC}}{g}. \]

But the time that the body takes to move in the parabolic path NT, is the same as its falling down through NC; hence

\[ CT = v \cdot t = \sqrt{2g \cdot BN} \cdot \sqrt{\frac{2CN}{g}} = 2\sqrt{BN \cdot NC}. \]

Now, if we describe a semicircle on SC as diameter, we have (by Eucl. III, 35) \( BN \cdot NC = NG^2 \); where \( NG \) is perpendicular to SC; wherefore \( CT = 2NG \); that is, the horizontal distance or range is twice GN, the ordinate of the semicircle.

210. Cor. 1.—The horizontal range will evidently be a maximum when the orifice is at the middle point Q of the diameter; and, therefore, the range will in this case be equal to the height of the vessel, or BC = CT.

211. Cor. 2.—If the orifice should be at \( n \), and \( Sn = NC \), then the jet will assume the path \( nT \), and its range will be CT, the same as when the orifice was at N.

212. Cor. 3.—If the orifice be at C, and the direction of projection be CG, inclined to the horizon at an angle of 45°, then we shall have the greatest range, CK, that the fluid vein can possibly take when projected under the charge SC.

213. Cor. 4.—If the direction of projection be CG, inclined at an angle of 75° to the horizon, its range will also be CK.

214. If the inclination be either less or more than 45° or 75° respectively, the ranges will be less than CK; resistances and all retarding causes whatsoever being neglected.

\( r \) and \( r' \) are called the greatest heights of the paths CrK, and CRK respectively, being the middle points of these paths.

215. Prop. XIX.—If water flow through a pipe which is kept constantly full, then the velocities of the fluid particles in different sections will vary inversely as the areas of these sections.

It is presumed that each particle in the same transverse section has the same velocity. Let then \( Q \) = volume of fluid passing through any portion of the pipe in one second, and \( v \) = the velocity of water in the same time while passing through that section of the pipe the area of which is \( a \).

Hence \( a \cdot v = Q \); \( \therefore v = \frac{Q}{a} \propto \frac{1}{a} \), since the volume passing through every second is constant.

216. Cor.—The volume which passes through the pipe in a time \( T = a \cdot v \cdot T \).

217. Prop. XX.—To determine the pressure exerted upon pipes by the water which flows through them.

Let us suppose the column of fluid CD divided into an infinite number of laminae EF, &c. Then friction being abstracted, every particle of each lamina will move with the same velocity when the pipe CD is horizontal. Now, the velocity at the *vena contracta*, \( sm \), may be expressed by \( \sqrt{A} \), \( A \) being the altitude of the fluid in the reservoir.

But the velocity at the *vena contracta* is to the velocity in the pipe as the area of the latter is to the area of the for- Motion of water. Therefore, \( \delta \) being the diameter of the rena con- fields, &c., tracts, and \( d \) that of the pipe CD, the area of the one will be to the area of the other, as \( \delta^2 : d^2 \) (Geometry, Sect. VI. Prop. IV.), consequently we shall have \( d^2 : \delta^2 = A : \) \( \frac{\delta}{d} \), the velocity of the water in the pipe. But since the velocity \( \sqrt{A} \) is due to the altitude \( A \), the velocity \( \frac{\delta}{d} \sqrt{A} \) will be due to the altitude \( \frac{\delta}{d} \). Now, as each particle of fluid which successively reaches the extremity DH of the pipe, has a tendency to move with the velocity \( \sqrt{A} \), while it moves only with the velocity \( \frac{\delta}{d} \sqrt{A} \), the ex- tremity DH of the pipe will sustain a pressure equal to the difference of the pressures produced by the velocities \( \sqrt{A} \) and \( \frac{\delta}{d} \sqrt{A} \), that is, by a pressure \( A - \frac{\delta}{d} \), A represent- ing the pressure which produces the velocity \( \sqrt{A} \), and \( \frac{\delta}{d} \) the pressure which produces the velocity \( \frac{\delta}{d} \). But this pressure is distributed through every part of the pipe CD, consequently the pressure sustained by the sides of the pipe will be \( A - \frac{\delta}{d} \).

218. Cor. 1.—If a very small aperture be made in the side of the pipe, the water will issue with a velocity due to the height \( A - \frac{\delta}{d} \). When the diameter \( \delta \) of the orifice is equal to the diameter \( d \) of the pipe, the altitude becomes \( A - \delta \) or nothing; and if the orifice is in this case below the pipe, the water will descend through it by drops. Hence we see the mistake of those who have maintained, that when a lateral orifice is pierced in the side of a pipe, the water will rise to a height due to the velocity of the included water.

219. Cor. 2.—Since the quantities of water, discharged by the same orifice, are proportional to the square roots of the altitudes of the reservoir, or to the pressures exerted at the orifice, the quantity of water discharged by a lateral ori- fice may be easily found. Let \( W \) be the quantity of water dis- charged in a given time by the proposed aperture under the pressure \( A \), and let \( w \) be the quantity discharged under the pressure \( A - \frac{\delta}{d} \). Then \( W : w = \sqrt{A} : \sqrt{A} - \frac{\delta}{d} \); consequently, \( w \times \sqrt{A} = W \times \sqrt{A} - \frac{\delta}{d} \), and \( w = \frac{W \times \sqrt{A} - \frac{\delta}{d}}{\sqrt{A}} = W \sqrt{\frac{d^2 - \delta^2}{d^2}} \). Therefore, since \( W \) may be determined by the experiments in the following chapter, \( w \) is known.

CHAPTER II.—ACCOUNT OF EXPERIMENTS FROM VESSELS, EITHER BY ORIFICES OR BY PIPES OR OPEN CANALS OR RIVERS.

220. We have already seen in our theoretical part (from 169 to 214), that the reservoir, out of which issues the water, may be always maintained at the same level; or, re- ceiving no supply, may be exhausted; or, lastly, the escape, instead of taking place in air, may do so into another reser- voir more or less full. These are the three principal cases to which experimenters on this subject have generally turned their attention.

Before detailing any experiments, let the following im- portant definitions be remembered:

221. Water may escape from a vessel, reservoir, or tank, by a horizontal opening in the bottom or in one of its late- ral sides; when the latter is the case, the opening is known as an orifice. The orifice is frequently said to be in a thin plate or wall, that is, in a wall the thickness of which is scarcely one-half the least dimension of the opening; it is also at times furnished with an adjutage, or short pipe, either cylindrical, or conical, converging towards, or diverg- ing from, the basin. It is clear that an orifice in a thin wall, but furnished with a pipe, would be equivalent to an orifice in a thick wall. When the upper border is away, and the issuing fluid is in contact with the containing vessel on three sides only, the orifice is then said to be a decoursoir, a notch, a weir, or waste-board; such are the waste-boards of canals and reservoirs, and weirs in rivers, which may be seen stretched across their course, so that the water being intercepted by them, must rise and flow over the crest or summit. A fluid vein in passing through a complete open- ing suffers a contraction, the phenomena of which will be fully explained in the following pages. The vertical dis- tance of the fluid surface in the vessel above the centre of gravity of the orifice, is called the height of the reservoir, or the charge of water over the orifice, or it is that charge under which the escape takes place.

The waste, or real effective waste of an orifice, is that vol- ume of fluid discharged in one unit of time, as a second. The theoretical waste is that volume per unit of time which theory assigns. The former is always less than the latter. The quotient or ratio of the real to the theoretical waste is termed the coefficient of contraction, or that fractional number which multiplies the theoretical to obtain the real waste.

222. Mariotte, 150 years ago, showed that in the case of fluids the velocities are as the square roots of the charges (Cor. 1. Prop. VIII.) Many succeeding experiments have proved the same principle, and although it is not rigidly true, it may still be admitted.

The following table of results will exhibit the propor- tionality. The charges are as 1 : 200 and more, and the section of the orifices are as 1 : 500, still in each the velocity has followed the square root of the charge; minute errors, causing small differences, arise, but these may be neglected. Such experiments have for their object the determination of the waste per unit of time; but it is clear that so long as the orifice remains of the same area the discharge or waste is exactly proportional to the velocity; and, therefore, a column denot- ing the relations of the one, must also give those of the other.

| Observer | Diameter of orifice in metres | Charge above orifice in metres | Series of Roots of charges | Waste or velocities | |-------------------|-------------------------------|-------------------------------|---------------------------|---------------------| | Daubisson and Castel | 0.01 | 0.025 | 1.000 | 1.000 | | | | 0.03 | 1.074 | 1.064 | | | | 0.04 | 1.241 | 1.244 | | | | 0.05 | 1.386 | 1.393 | | | | 0.06 | 1.531 | 1.524 | | | | 1.0 | 3.000 | 3.000 | | Bossut | 0.027 | 2.22 | 1.500 | 1.497 | | | | 3.81 | 1.713 | 1.707 | | | | 2.94 | 1.000 | 1.000 | | Michelotti | 0.081 | 3.81 | 1.305 | 1.301 | | | | 6.76 | 1.738 | 1.692 | | | | 2.11 | 1.000 | 1.000 | | | | 3.66 | 1.316 | 1.315 | | | | 0.40 | 1.000 | 1.000 | | Poncelet and Lesbros | Square of 20 centi- | 0.70 | 1.323 | 1.330 | | | | metres | 1.00 | 1.000 | | | | 1.30 | 1.803 | 1.808 | | | | 1.60 | 2.000 | 2.000 | 223. It has been shown by experiment that the principle holds for all fluids whatever, as mercury, oils, &c., and even for aeriform gases; so that any one of them issuing from an orifice will do so with a velocity independent of the nature and density of the substance; it will only depend on the charge. The same principle has also been detected in the case of discharges in canals, as well as in the atmosphere; the velocity will always be the same with the same head, whatever be the pressure on the free surface of the fluid, provided only that the jet at the orifice is subjected to an equal exterior pressure. But if these pressures be unequal, then the velocities differ.

224. We have seen (in 173) the relation of the several parts of the contracted vein to each other; it may be interesting to know the values which the older experimenters gave for the same parts. Newton concluded that the ratio of the contracted section \(op\) (fig. 43), and the area of the orifice \(mn\), was \(1 : \sqrt{2}\), and consequently that the diameters of these sections respectively were as \(0.841 : 1.0\). If \(mn\) be 1, Poleni found that \(op\) would be 0.79; Borda, 0.804; Michelotti, 0.792; Bossut, from 0.812 to 0.817; Eytelwein, 0.80; Venturi, 0.798; Brunacci, 0.78. The mean of all these values for the same thing, \(op\), is 0.80. But from a mean of more recent experiments by Ignazio Michelotti, the son of F. D. Michelotti, we have the relation of 1 to 0.787 as the ratio of the area of the orifice and that of the contracted section. Since the latter experiments are regarded as very accurate, Daubisson has followed the ratio 1 : 0.787.

225. Were the velocity at the contracted section really that due to the height of the reservoir, and were the escape to take place through an adjuage of the same form as the contracted vein, then on introducing into the expression for the waste the area \(S\) of the exterior orifice of this adjuage, the calculated would be equal to the actual waste, and the ratio of the coefficients would be 1. Michelotti, in one of his experiments, employed a cycloidal adjuage, and obtained a coefficient of 0.84. In all probability he would have arrived at 1, if the walls of the adjuage had been more accurately adapted to the form of the fluid vein, and if the resistance of these walls, and also that of the air, had not somewhat retarded the motion.

226. The coefficient, or the ratio mentioned above, is not always a constant quantity. It varies with the form and position of the orifice, with the thickness of the plate in which the orifice is made, as well as the form of the vessel and the weight of the superincumbent fluid. This variable quantity, therefore, must not be neglected in practice.

SECTION I.—ON THE VOLUME OF WATER DISCHARGED FROM VESSELS KEPT CONSTANTLY FULL, BY ORIFICES IN THIN PLATES.

(a.) Flow of Water through an Orifice in a Thin Plate.

227. We proceed now to determine directly that fractional number, or coefficient, which will enable us to reduce the theoretical to the actual waste. As this is a mere tentative process, we must gauge with care the volume of water flowing out of a given orifice, under a constant charge, and in a given time, from which is found the volume in one second, or the real waste for that time; on dividing this by the volume which theory gives as issuing from the same orifice, and under the same circumstances, we find the coefficient required. Thus, let the orifice be 3 inches in diameter, then the head of 4 feet of water issues out of it with a velocity of 16 feet per second. Now, area of orifice \(= \pi r^2 = (3.1416)(1.5925)^2 = 7.97\) square inches, which, if multiplied into the velocity of each particle of water, or 4.012, gives the volume of a Motion of prism or cylinder equal to that of the water discharged, or 7.97 cubic feet per second. But the real discharge in 1 minute was found to be 49.68 cubic feet, which is at the rate of 0.552 cubic feet per second; hence, coefficient \(= \frac{0.552}{0.8894} = 0.620\). A considerable number of experimental philosophers since the time of Newton have endeavoured to determine its true value. Daubisson, in his Traité d'Hydraulique à l'usage des Ingénieurs, 1840, gives a table containing the principal results which several eminent experimenters have arrived at. They include circular, square, and rectangular orifices:

| Circular Orifices | Square Orifices | |------------------|----------------| | Observer | Diameter | Charge | Coefficient | Observer | Side of Square | Charge | Coefficient | | Mariette | 0.008 | 0.78 | 0.029 | Castel | 0.01 | 0.655 | | Da | 0.008 | 0.90 | 0.026 | Boussat | 0.027 | 0.616 | | Castel | 0.01 | 0.65 | 0.073 | Michelotti| 0.027 | 0.607 | | Da | 0.015 | 0.31 | 0.054 | Boussat | 0.027 | 0.616 | | Castel | 0.015 | 0.138 | 0.054 | Boussat | 0.027 | 0.616 | | Da | 0.021 | 0.177 | 0.054 | Boussat | 0.027 | 0.616 | | Eytelwein | 0.021 | 0.232 | 0.054 | Boussat | 0.027 | 0.616 | | Bossut | 0.021 | 1.30 | 0.054 | Boussat | 0.027 | 0.616 | | Michelotti | 0.021 | 0.223 | 0.054 | Boussat | 0.027 | 0.616 | | Castel | 0.03 | 0.168 | 0.054 | Boussat | 0.027 | 0.616 | | Venturi | 0.03 | 0.222 | 0.054 | Boussat | 0.027 | 0.616 | | Bossut | 0.03 | 0.381 | 0.054 | Boussat | 0.027 | 0.616 | | Michelotti | 0.03 | 0.220 | 0.054 | Boussat | 0.027 | 0.616 | | Da | 0.03 | 0.224 | 0.054 | Boussat | 0.027 | 0.616 | | Da | 0.03 | 0.381 | 0.054 | Boussat | 0.027 | 0.616 | | Da | 0.03 | 0.76 | 0.054 | Boussat | 0.027 | 0.616 | | Da | 0.03 | 2.11 | 0.054 | Boussat | 0.027 | 0.616 | | Da | 0.03 | 3.66 | 0.054 | Boussat | 0.027 | 0.616 |

The most remarkable of these experiments, of which the above table is the result, were those of the elder Michelotti, performed in the year 1764, at about 3 miles from Turin. The internal dimension of the reservoir was a square of 0.97 met., and 8 met. in height; it was supplied with water from the River Dora by a canal of derivation. On one of the faces of this reservoir was arranged a series of adjuages at different, yet convenient depths; on the surface of the ground, also, were arranged different vessels for measuring the actual discharges. Ignazio Michelotti performed these same experiments in 1784, the results of which—the five last—are tabulated as above. It will be observed from the table that the coefficients from the large orifices are higher than the others, although contrary to the rule that would be deduced from the experiments in general; some particular circumstance must have produced this anomaly. With respect to the experiments of M. Castel, engineer to the Water-works of Toulouse, they were carried on by him in company with M. Daubisson at Toulouse, and in spite of all the care which they took in their operations, the smallness of the orifices would only permit them to come within \( \pm \frac{1}{10} \)th of the value of the coefficient. They principally occupied themselves on the orifice 0.01 met., as being in a manner the starting point in a distribution of water, made after the metrical system of weights and measures.

228. In the year 1782, the engineer Espinasé made several experiments in the canal of Languedoc, in order to determine the coefficient when water issues from a sluice-gate. The width of the sluice was 1.3 met., the area

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1 Mémoire Physico-Mathématique contenant les résultats d'expériences Hydrauliques. Dans les Mémoires de l'Académie des Sciences de Turin, 1784-1785. 2 Sperimenti Idraulici, &c., de F. D. Michelotti, Torino, 1767 et 1771. Motion of the opening varied from 0.7194 to 0.6240 sq. met., and the height from 0.55 to 0.48 met., the charge over the centre being from 4.36 to 1.975 met., and the form of the orifice was not exactly rectangular; the mean results of all the coefficients was 0.625, which has been regarded as large; but it is accounted for from the fact that the escape did not take place in a thin wall or plate, and that the contraction on some point was suppressed; the wood work around the opening of the sluice was 0.27 met. thick, and on the lower edge it was even 0.54 met. Further, the coefficient, which was 0.641 met. for an opening of 0.46 met., rose to 0.803 met. when the paddle over the opening was raised 0.12 met. Six years after, M. Pin, engineer on the same canal, performed a series of similar experiments, with similar results, and which may be seen in the Anciens Mémoires de l'Académie des Sciences de Toulouse, tom. ii., 1784; see also Histoire du Canal du Midi ou de Languedoc, par le Général Androissy, tom. i., p. 251. A coefficient of the above value was found from an experiment on the sluice-gate of the basin of Havre (see Architecture Hydraulique, par Belidor et Navier, tom. i., p. 289).

229. But the most remarkable fact connected with the experiments of Lespinasse and Pin, is that when two equal openings are made, one in each of the wings of a sluice-gate, the waste of the first diminishes on the second commencing to flow, and the combined volume of discharge for both orifices during a certain time, is less than double the volume escaping by one of the openings in half that time; consequently, also, the coefficient for each case must be different, for different discharges are given. The distance between the openings was 2.92 met., and their plane formed an angle of 60° with the direction of the canal. The difference in the discharges was about one-eighth, and the difference in the coefficients will be seen from the annexed table.

This was regarded as true till 1836, when Daubisson and Castel showed its falsity. They had, from one side and another of a sluice-gate, three rectangular orifices of 0.10 met. in width, and 0.01 met. in height, separated by 0.01 met. The volume discharged by the middle orifice was first obtained, then the volume from two, and lastly, the discharge from the whole three. The mean results are contained in the following table:

| Charge on orifice | Waste of middle orifice | |------------------|------------------------| | Middle single opening | Middle with one side opening | | Middle with two side openings | Coefficient | | Met. | Cub. met. | Cub. met. | Cub. met. | | 0.02 | 0.000455 | 0.000455 | 0.000457 | 0.728 | | 0.03 | 0.000551 | 0.000551 | 0.000550 | 0.720 | | 0.04 | 0.000635 | 0.000635 | 0.000637 | 0.719 | | 0.05 | 0.000707 | 0.000707 | 0.000715 | 0.715 | | 0.06 | 0.000771 | 0.000769 | 0.000770 | 0.710 |

As these resulting coefficients were considerably larger than those of Lespinasse and Pin, Daubisson and Castel thought that the difference was owing to the too small distance separating one orifice from another; this difference was now made five times greater, and yet the coefficients remained constant. As the difference was still matter of surprise, M. Daubisson, in 1836, requested M. Castel to undertake a series of experiments on a more extensive scale than they had previously done. For this purpose M. Castel stopped up a canal in width 0.74 met. by a thin copper plate, in which he opened three rectangular orifices on the same horizontal base, each 0.10 met. wide, and 0.06 high, and the distance betwixt each was 0.08 met. The escape was produced under a constant charge of 0.107 met. above the centre, and the coefficients of contraction were as under:

- One orifice open: 0.6198 - Two orifices open: 0.6205 - The whole three orifices open: 0.6230

We here see that as more of the orifices were opened, the coefficients increased instead of diminished in value. As it belongs to a particular cause, a greater velocity of water in the canal, followed by a greater waste, M. Castel inferred that although new orifices should be made by the side of an already existing orifice, whether in a sluice-gate, reservoir, or stopped-up course of water, no diminution of waste will take place on this account.

230. It was supposed, however, that although this was shown to hold good in the case of orifices in a plane wall or plate, yet a difference would be found in those of planes making a certain angle, as in the case of flood-gates. M. Castel has also fully resolved this question. He joined two planes, making an angle of 120°, a flood-gate being 10° more open, and in each he made two rectangular orifices of 0.10 by 0.06 met.—one at 0.12, the other at 0.28 met. distance from the plane of junction; this kind of barrier was erected at the extremity of a canal, and the escape took place under a charge of 0.14 met. Each of the orifices was successively opened at first; then two at a time, combining them in different ways; then three similarly combined; and last of all, the whole four. The annexed table shows the mean results.

M. Castel also adapted on his experimental apparatus, described in 233, two orifices of 0.05 by 0.03 met., and found the following results, from which it will be seen that the same coefficient is always found with an insignificant augmentation, due to the number of open orifices.

231. The following table contains the results obtained by Michelotti when the orifices were vertical, and of a square or circular form, the altitude of the head of water varying from 6 to about 22 feet:

| Height of water above centre of orifice | Size and form of orifice | Time of flow | Volume discharged | |----------------------------------------|--------------------------|--------------|------------------| | Ft. in. lbs. pts. | Minutes | Ft. in. lbs. | | 6 7 4 3 | square of 3 inches | 10 | 463 | 7 3 | | 11 8 1 6 | | 8 | 516 | 9 5 | | 21 8 3 6 | square of 2 | 5 | 415 | 5 3 | | 6 7 6 0 | inches | 15 | 329 | 9 8 | | 21 5 3 7 | square of 1 | 10 | 324 | 9 0 | | 6 9 1 0 | inch | 60 | 158 | 6 7 | | 6 8 4 0 | circular, 3 inches | 60 | 562 | 11 4 | | 21 7 4 0 | diameter | 15 | 542 | 10 6 | | 6 9 5 0 | circular, 2 inches | 8 | 521 | 3 7 | | 21 10 10 0 | diameter | 30 | 488 | 8 3 | | 6 10 6 0 | circular, 1 inch | 20 | 575 | 5 10 | | 22 0 2 0 | diameter | 60 | 247 | 4 3 | | | | 60 | 444 | 6 5 |

The mean coefficient from Michelotti's experiments is 0.625.

232. Messrs Brindley and Smeaton found that 20 cubic feet of water were discharged from orifices 1 inch square, in the following times, varying with the height of the water:

| Height of water in feet | Time of discharging in seconds | |-------------------------|-------------------------------| | 1 | 562 | | 3 | 320 | | 5 | 234 |

When the height of the water was 6 feet, and the orifice an inch square, 20 cubic feet were discharged in 17 minutes 33 seconds. The coefficient obtained from these experiments is 0.63.

233. In the following experiments by the Abbé Bossut, which were frequently repeated, the orifice was pierced in a plate of copper about half a line thick:

Quantity of Water discharged in one minute, by Orifices differing in form and position.

| Altitude of the fluid above the centre of the orifice | Form and position of the orifice | The orifice's diameter | No. of cubic feet discharged in a minute | |------------------------------------------------------|---------------------------------|-----------------------|----------------------------------------| | Fr. In. Lin. | | | | | 11 8 10 | Circular and Horizontal | 1 inch | 9281 | | | Circular and Horizontal | 2 inches | 37203 | | | Horizontal and Square | 1 inch, side | 11817 | | | Horizontal and Square | 2 inch, side | 47351 | | 9 0 0 | Lateral and Circular | 6 lines | 2016 | | | Lateral and Circular | 1 inch | 8135 | | | Lateral and Circular | 6 lines | 1353 | | | Lateral and Circular | 1 inch | 6436 |

234. From the results contained in the preceding table, we may draw the following conclusions:

1. That the quantities of water discharged in equal times by different apertures, the altitudes of the fluid being the same, are very nearly as the areas of the orifices. That is, if $A$ and $a$ represent the areas of the orifices, and $W$, $w$ the quantities of water discharged,

$$ W : w = A : a. $$

2. The quantities discharged in equal times by the same aperture, the altitude of the fluid being different, are to one another very nearly as the square roots of the altitudes of the water in the reservoir, reckoning from the centres of the orifices. That is, if $H$, $h$ be the different altitudes of the fluid, we shall have

$$ W : w = \sqrt{H} : \sqrt{h}. $$

3. Hence we may conclude in general, that the quantities discharged in the same time by different apertures, and under different altitudes in the reservoir, are in the compound ratio of the areas of the orifices, and the square roots of the altitudes. Thus, if $W$, $w$ be the quantities discharged in the same time from the orifices $A$, $a$, under the same altitude of water; and if $W'$, $w'$ be the quantities discharged in the same time by the same aperture $a$, under different altitudes, $H$, $h$; then, by the first of the two preceding articles,

$$ W : w = A : a, $$

and by the second

$$ w : w' = \sqrt{H} : \sqrt{h}. $$

Multiplying these analogies together, gives us

$$ W : W' = A\sqrt{H} : a\sqrt{h}, $$

and by dividing by $w$,

$$ W : W' = A\sqrt{H} : a\sqrt{h}. $$

This rule is sufficiently correct in practice; but when great accuracy is required, the following remarks must be attended to.

4. Small orifices discharge less water in proportion than great ones, the altitude of the fluid being the same. The circumferences of the small orifices being greater in proportion to the issuing column of fluid than the circumferences of greater ones, the friction, which increases with the area of the rubbing surfaces, will also be greater, and will therefore diminish the velocity, and consequently the quantity discharged.

5. Hence of several orifices whose areas are equal, that which has the smallest circumference will discharge more water than the rest under the same altitude of fluid in the reservoir, because in this case the friction will be least. Circular orifices, therefore, are the most advantageous of all, for the circumference of a circle is the shortest of all lines that can be employed to inclose a given space.

6. In consequence of a small increase which the contraction of the vein of fluid undergoes, in proportion as the altitude of the water in the reservoir augment, the quantity discharged ought also to diminish a little as that altitude increases.

By attending to the preceding observations, the results of theory may be so corrected, that the quantities of water discharged in a given time may be determined with the greatest accuracy possible.

235. The Abbé Bossut has given the following table, containing a comparison of the theoretical with the real discharges, for an orifice one inch in diameter, and for different altitudes of the fluid in the reservoir. The real discharges were not found immediately by experiment, but were determined by the precautions pointed out in the preceding articles, and may be regarded to be as accurate as if direct experiments had been employed:

Comparison of the Theoretic with the Real Discharges from an Orifice one inch in diameter (from Prony's Arch. Hydraul. l. 369).

| Constant altitude of the water in the reservoir above the centre of the orifice | Theoretical discharges through a circular orifice one inch in diameter | Real discharges in the same time through the same orifice | Ratio of the theoretical to the real discharges | |-----------------------------------------------------------------------------|------------------------------------------------------------------|----------------------------------------------------------|-----------------------------------------------| | Paris feet. | Cubic inches. | Cubic inches. | | | 1 | 4381 | 2722 | 1 to 0.62133 | | 3 | 7559 | 4710 | 1 to 0.62064 | | 5 | 9797 | 6075 | 1 to 0.62010 | | 7 | 11592 | 7183 | 1 to 0.61985 | | 9 | 13144 | 8135 | 1 to 0.61892 | | 11 | 14530 | 8990 | 1 to 0.61873 | | 13 | 15797 | 9764 | 1 to 0.61810 | | 15 | 16968 | 10472 | 1 to 0.61716 |

236. It is evident from the preceding table, that the theoretical, as well as the real discharges, are nearly proportional to the square roots of the altitudes of the fluid in the reservoir.

The fourth column of the preceding table also shows us that the theoretical are to the real discharges nearly in the ratio of 1 to 0.62, or more accurately, as 1 to 0.61938; therefore 0.62 is the number by which we must multiply the discharges as found by the formulae in the preceding chapter, in order to have the quantities of water actually discharged.

237. In order to find the quantities of fluid discharged by orifices of different sizes, and under different altitudes of water in the reservoir, we must use the table in the following manner:—Let it be required, for example, to find the quantity of water furnished by an orifice three inches in diameter, the altitude of the water in the reservoir being 30 feet. As the real discharges are in the compound ratio of the area of the orifices, and the square roots of the altitudes of the fluid (art. 234, No. 3), and as the theoretical quantity of water discharged by an orifice one inch in diameter, is, by the second column of the table, 16968 cubic inches in a minute, we shall have this analogy,—$1/\sqrt{15}:9/\sqrt{30}=16968:215961$ cubic inches, the quantity required. This quantity being diminished in the ratio of 1 to 0.62, being the ratio of the theoretical to the actual discharges, gives 133896 for the real quantity of water discharged by the given orifice. But (by No. 5 of art. 234), the quantity discharged ought to be a little greater than 133896, because greater orifices discharge more than small ones; and by No. 6 the quantity ought to be less than 133896, because the altitude of the fluid is double that in the table. These two causes therefore having a tendency to increase and diminish the quantity deduced from the preceding table, we may regard 133896 as very near the truth. Had the orifice been less than one inch, or the altitude less than 15 feet, it would have been necessary to diminish the preceding answer by a few cubic inches. Since the velocities of the issuing fluid are as the quantities discharged, the preceding results may be employed also to find the real velocities from those which are deduced from theory.

238. As the velocity of falling bodies is $16087$ feet per second, the velocity due to $16087$ feet will be $32174$ feet per second; and as the velocities are as the square roots of the height, we shall have $\sqrt{16087} : \sqrt{H} = 32174 : V$, the velocity due to any other height; consequently $V = \frac{32174\sqrt{H}}{\sqrt{16087}} = \frac{32174\sqrt{H}}{4011} = 8.016VH$, so that $8.016$ is the coefficient by which we must always multiply the altitude of the fluid in order to have its theoretical velocity.

We have already seen that the jet takes a parabolic path, and that the abscissae and ordinates of its various points can be determined (177). Both Bossut and Michelotti took special care in obtaining true values for these lines so as to find the value of the height due to the velocity of exit ($h = \frac{v^2}{4g}$) (177); and for this velocity we have $v = \sqrt{2gh}$: from which relations we may compare the real with the theoretical velocity, and $h$ with $H$ the charge. The following table from Michelotti's experiments will let us see the near coincidence of these quantities:

| Curve of Jet | Height due to velocity of exit | Charge | Real | Theoretical | |--------------|-------------------------------|--------|------|------------| | Abscissa | Ordinate | | | | | Met. | Met. | Min. | Min. | | | 6-28 | 7-53 | 2-257 | 2-29 | 0-65 | | 4-66 | 8-45 | 3-83 | 3-93 | 0-87 | | 1-41 | 6-25 | 6-92 | 7-19 | 11-67 |

From the above it appears that the difference between the third and fourth columns increases with the charge, the cause of this difference being the resistance of the air increasing as the square of the velocity, and consequently nearly as the charge. Were it not for this, the difference would be very nearly equal to zero, and the velocity at the contracted section would have been equal to that due to the charge. Hence it may be said that water escaping through an orifice in a thin plate issues with a velocity which is very nearly equal to that due to the height of the reservoir or charge, and is not sensibly diminished by the contraction.

SECTION II.—ON THE QUANTITY OF WATER DISCHARGED FROM VESSELS CONSTANTLY FULL, BY SMALL TUBES ADAPTED TO CYLINDRICAL ORIFICES.

(β.) Flow through Cylindrical Adjutages.

239. The difference between the actual discharges, and those deduced from theory, arises from the contraction of the fluid vein, and from the friction of the water against the circumference of the orifice. If the operation of any of these causes could be prevented, the quantities of water actually discharged would approach nearer the theoretical discharges. There is no probability of diminishing friction in the present case by the application of ungents; but, if a short cylindrical tube be inserted in the orifice of the vessel, the water will follow the sides of the tube, the contraction of the fluid vein will be in a great measure prevented, and the actual discharges will approximate much nearer to those deduced from theory than when the fluid issues through a simple orifice.

240. If a cylindrical tube, 2 inches long and 2 inches in diameter, be inserted in the reservoir, and if this orifice be stopped by a piston till the reservoir is filled with water, the fluid, when permitted to escape, will not follow the sides of the tube, that is, the tube will not be filled with water, and the contraction in the vein of fluid will take place in the same manner as if the orifice were pierced in a thin plate. When the cylindrical tube was 1 inch in diameter and 2 inches long, the water followed the sides of the tube, and the vein of fluid ceased to contract. While M. Bossut was repeating this experiment, he prevented the escape of the fluid by placing the instrument MN (see fig. 55), consisting of a handle and a circular head, upon the interior extremity of the tube, and found, to his great surprise, that when he withdrew the instrument MN, to give passage to the water, it sometimes followed the sides of the tube, and sometimes detached itself from them, and produced a contraction in the fluid vein similar to that which took place when the first tube was employed. After a little practice, he could produce either of these effects at pleasure. The same phenomenon was exhibited when the length of the tube was diminished to 1 inch 6 lines; only it was more difficult to make the fluid follow the circumference of the tube. This effect was still more difficult to produce when its length was reduced to 1 inch; and when it was so small as half an inch, the water uniformly detached itself from its circumference, and formed the rena contracta.

241. The coefficient of reduction of the theoretical to the actual waste, with such small tubes, will be found in the following table. It will be seen that the additional tube, that is, a cylindrical adjutage, does not cause much variation:

| Observer | Diameter | Length | Charge | Coefficient | |----------|----------|--------|--------|-------------| | Castel | 0-0155 | 0-040 | 0-20 | 0-827 | | Do | 0-0155 | 0-040 | 0-48 | 0-829 | | Do | 0-0155 | 0-040 | 0-99 | 0-829 | | Do | 0-0155 | 0-040 | 2-00 | 0-829 | | Do | 0-0155 | 0-040 | 3-03 | 0-830 | | Bossut | 0-023 | 0-054 | 0-65 | 0-788 | | Do | 0-023 | 0-054 | 1-24 | 0-787 | | Eythelwein| 0-026 | 0-078 | 0-72 | 0-821 | | Bossut | 0-027 | 0-041 | 3-85 | 0-804 | | Do | 0-027 | 0-054 | 3-87 | 0-804 | | Do | 0-027 | 0-108 | 3-82 | 0-804 | | Ventral | 0-041 | 0-128 | 0-88 | 0-822 | | Michelotti| 0-081 | 0-216 | 2-18 | 0-815 | | Square | 0-081 | 0-216 | 3-89 | 0-803 | | Do | 0-081 | 0-216 | 6-71 | 0-803 |

Bossut's two first observations are looked upon as anomalous, but the mean of all the others is 0-817, and may be regarded as 0-82.

242. Quantities of Water discharged by Cylindrical Tubes one inch in diameter with different lengths.

| Constant altitude of the fluid above the superior edge of the tube being 11 feet 8 inches and 10 lines. | Variable length of the tubes expressed in lines. | Cubic inches discharged in a minute. | |-------------------------------------------------------------------------------------------------|--------------------------------------------------|-------------------------------------| | The tube being filled with the issuing fluid. | 48 | 12274 | | | 24 | 12188 | | | 18 | 12108 | | The tube not filled with the issuing fluid. | 18 | 9222 |

The experiments in the preceding table were made with tubes inserted in the bottom of the vessel. When the tubes were fixed horizontally in the side of the reservoir, they furnished the very same quantities of fluid, their dimensions and the altitude of the fluid remaining the same. It appears, from the preceding results, that the quantities of water discharged increase with the length of the tube, and that these quantities are very nearly as the square roots of the altitudes of the fluid above the interior orifice of the vertical tube.

We have already seen that the theoretical are to the real discharges as 1 to 0.62, or nearly as 16:1 to 10. But, by comparing the two last experiments in the preceding table, it appears that the quantity of fluid discharged by a cylindrical tube where the water follows its sides, is to the quantity discharged by the same tube when the rena contracta is formed, as 13 to 10; and since the same quantity must be discharged by the latter method as by a simple orifice, we may conclude that the quantity discharged according to theory, and that which is discharged by a cylindrical tube and by a simple orifice, are to one another very nearly as the numbers 16, 13, 10. Though the water, therefore, follows the sides of the cylindrical tube, the contraction of the fluid vein is not wholly destroyed; for the difference between the quantity discharged in this case, and that deduced from theory, is too great to be ascribed to the increase of friction which arises from the water following the circumference of the tube.

243. In order to determine the effects of tubes of different diameters, under different altitudes of water in the reservoir, M. Bossut instituted the experiments the results of which are exhibited in the following table:

Quantities of Water discharged by Cylindrical Tubes two inches long, with different Diameters.

| Constant altitude of the water above the orifice | Diameter of the Tube | Quantity of water discharged in a minute. | |--------------------------------------------------|----------------------|------------------------------------------| | Feet. Inches. | Lines. | Cubic inches | | | | | | | | | | | | |

244. By comparing the different numbers in this table, we may conclude—

1. That the quantities of water discharged by different cylindrical tubes of the same length, the altitude of the fluid remaining the same, are nearly as the areas of the orifices, or the squares of their diameters.

2. That the quantities discharged by cylindrical tubes of the same diameter and length are nearly as the square roots of the altitude of the fluid in the reservoir.

3. Hence the quantities discharged during the same time, by tubes of different diameters, under different altitudes of fluid in the reservoir, are nearly in the compound ratio of the squares of the diameters of the tube, and the square roots of the altitudes of the water in the reservoir.

4. By comparing these results with those which were deduced from the experiments with simple orifices, it will be seen that the discharges follow the same laws in cylindrical tubes as in simple orifices.

245. The following table is deduced from the foregoing experiments, and contains a comparative view of the quantities of water discharged by a simple orifice, according to theory, and those discharged by a cylindrical tube of the same diameter under different altitudes of water. The numbers might have been more accurate by attending to some of the preceding remarks; but they are sufficiently exact for any practical purpose. The fourth column, containing the ratio between the theoretical and actual discharges, was computed by M. Prony:

Comparison of the Theoretical with the Real Discharges from a Cylindrical Tube one inch in Diameter, and two inches long.

| Constant altitude of the water in the reservoir above the orifice | Theoretical discharge through a circular orifice of the same diameter. | Real discharge in the same time by a cylindrical tube one inch diameter, and two inches long. | Ratio of the theoretical to the actual discharge. | |-------------------------------------------------------------------|---------------------------------------------------------------------|---------------------------------------------------------------------------------|--------------------------------------------------| | Feet. Inches. | Cubic inches | Cubic inches | Ratio | | | | | | | | | | | | | | | |

By comparing the preceding table with that in art. 235, we shall find that cylindrical tubes discharge a much greater quantity of water than simple orifices of the same diameter, and that the quantities discharged are as 81 to 62 nearly. This is a curious phenomenon, and will be afterwards explained.

246. The application of this table to other additional tubes under different altitudes of the fluid not contained in the first column, is very simple. Let it be required, for example, to find the quantity of water discharged by a cylindrical tube 4 inches in diameter and 8 inches long, the altitude of the fluid in the reservoir being 25 feet. In order to resolve this question, find (by art. 237) the theoretical quantity discharged, which in the present instance will be 350490 cubic inches, and this number diminished in the ratio of 1 to 0.81 will give 284773 for the quantity required. The length of the tube in this example was made 8 inches, because, when the length of the tube is less than twice its diameter, the water does not easily follow its interior circumference. If the tube were longer than 8 inches, the quantity of fluid discharged would have been greater, because it uniformly increases with the length of the tube; the greatest length of the tube being always small, in comparison with the altitude of the fluid in the reservoir.

247. When the cylindrical tube is two inches long, and its diameter one, the water will fill the tube, and issue in threads parallel to the axis of the orifice; the vein also will cease to contract, and the diminution of the discharge can only take place from a diminution of the velocity; the ratio, then, of the actual to the theoretic discharge will be the same as that of the actual to the theoretic velocity. The following table lets us see this:

| Observer | Jet. | Velocity. | Coefficient | |----------|------|-----------|-------------| | | Absciss. | Ordin. | Real. | Theoretic. | Of Veloc. | Of Waste. | | Venturi | 1462 | 1898 | 3415 | 1544 | 0.824 | 0.822 | | Castel | 9446 | 673 | 4017 | 2426 | 0.832 | 0.827 | | Castel | 1140 | 1769 | 3669 | 4414 | 0.832 | 0.829 |

From the above table we see that the coefficients of discharge and velocity are sensibly equal. It appears, also, that the velocity of a jet issuing from a cylindrical adjutage is equal to 0.82 of that due to the charge or height of the reservoir; and that the height due to the velocity of exit is only \( \frac{0.82}{0.82} = 0.6724 \) of the last; since the heights of the charges are as the squares of the velocities.

248. The coefficient, it will be perceived, has risen from 0.62 to 0.82, the cause of which, together with the increased... waste, are probably due to the following circumstances.

The fluid vein, after its contraction, on entering the additional adjutage, tends to assume and preserve a cylindrical form, the section of which will be that of the contracted vein, and consequently it tends to issue without touching the walls of the cylindrical tube. Some fillet threads, however, move towards these walls, owing either to some divergence in their direction, or to some attractive force, or both combined. When a contact of fluid particles takes place at the walls, then they are strongly held by molecular attraction; and the same force causes particle to cling to particle, even over the whole vein, which issues from a full pipe, and passes quickly by the contracted section. The immediate cause, however, is contact, and every circumstance which favours that will tend to produce an augmentation in the value of the coefficient and volume discharged.

The principal of these circumstances are:

1. The length of the tube; the longer it is the greater is the tendency to contact; the less it is the less is that tendency.

2. A feeble velocity; the fluid threads will then be drawn with greater facility towards the walls.

3. The affinity of the matter of the tube for the fluid, or rather its disposition to be more easily wetted.

249. Venturi made several ingenious experiments on cylindrical adjutages, piercing the thin plate of a reservoir, the orifice of which was in diameter 0·0406 met.; and under a charge of 0·88 met., he obtained a volume of 0·137 cub. met. of water in 41 seconds. To this simple orifice he next adapted a cylindrical pipe, not continued beyond the narrowest part of the vein, and having a form very nearly the same; the diameter of the contraction being 0·0327 met., and its distance from the orifice 0·025; and under the same charge he obtained the same volume, but now in 42 seconds. To this first pipe he added a continuation beyond the contracted part, of the same dimensions as the orifice, and to obtain the same volume under the same head, the time required was only 31 seconds. Lastly, to the orifice he fitted on a uniform pipe of the same length as in last experiment, and the volume escaped in 31 seconds was 0·137 cub. met.

It appears, then, that the introduction of a cylindrical adjutage does not diminish the contraction, but makes the fluid pass by the contracted section with a greater velocity in the ratio of 31 : 41 or 42; and hence the augmentation of the waste. Venturi attributed the cause of the above to an excess of the atmospheric pressure on the fluid surface contained in the reservoir, an excess arising from a void in that part of the adjutage where the greatest contraction takes place. He endeavoured to prove this by many very interesting experiments, but his results have been too much generalized.

Two of the most beautiful experiments of Venturi are explanatory of what he termed the lateral communication of motion in fluids.

(v.) Flow through Converging Conical Adjutages.

250. The investigations on pipes of this kind were up to 1838 very inexact and imperfect. Four experiments made on this subject by the Marquis Poleni, are recorded in his De Castellis per quae derivantur Fluxiorum Aqua (1718), and cited by Bossut in his Hydrodynamique. Their accuracy, however, was doubted by Daubisson, and this circumstance, together with the paucity of results on converging conical adjutages, determined him, along with Castel, to project a series of experiments which we shall notice presently. With respect, however, to very large conical adjutages, or rather pyramidal nozzles, employed for the purpose of putting hydraulic wheels in motion (458), we have three very accurate experiments of the engineer Lespinasse on the mills of the canal of Languedoc. The nozzles in these experiments were frusta of a rectangular pyramid, having a length of 2·923 met., the greater base 0·731 by 0·975 met., the smaller base 0·135 by 0·190 met. The opposite faces were respectively inclined at angles of 11° 38' and 15° 18', and the charge was 2·923 met. The two first experiments, of which the results are here given in a tabular form, were made on a mill with two millstones, furnished each with a separate wheel. In the first experiment the water was only given to a single wheel, but in the second it was given to both at once. Such adjutages were found to diminish the waste very little, the real differing from the theoretic waste by between 100th and 200ths of the latter.

251. As in this adjutage there are two contractions (171), the formula given in Prop. II. (181), to calculate the waste for a single contraction is inapplicable in the present case. Let \( n \) be the exterior contraction, or the ratio of the contracted section of the fluid to that of the orifice, and \( n' \) the coefficient of the velocity, or the ratio of the real to the theoretic velocity, \( n' \) will be the coefficient of the waste, or the ratio of the real to the theoretic waste. Let \( S \) represent the sectional area of the orifice, \( V \) the velocity due to the charge, the real waste will be expressed by the formula \( nS^2n'V = nS\sqrt{2gH} \) (Prop. II. 181). The quantities \( n \) and \( n' \) are found by experiment. The coefficients of velocity and discharge find a very important application in the case of jets of water, as in fire-engines (517) and fountains.

252. To determine these coefficients, and specially to determine the angle of convergence which would cause the greatest discharge, a variety of adjutages were submitted to experiment by M. Castel, in each of which the diameter of the orifice of issue and the length of adjutage remained constant; but the diameter of entrance, and consequently the angle of convergence, gradually increased. The flow was produced under different charges for each adjutage. At every experiment the real waste was determined by direct gauging, and the velocity of escape by the method of the parabola (Prop. I. 177). The waste divided by \( SV \) gave \( n' \); and the observed velocity divided by \( V(\sqrt{2gH}) \) gave \( n' \). The series of numbers for \( n' \) showed the waste corresponding to each angle of convergence, and consequently the angle of maximum waste; and the series of numbers for \( n' \) indicated the progression by which the velocities increased.

253. So early as 1831, M. Castel had, with a very small apparatus, and under feeble charges, made a series of experiments on adjutages of this nature. The results were detailed in the Annales des Mines for 1833. In 1837 he renewed his experiments, and considerably extended his labours by means of the beautiful experimental apparatus at the Water-house of Toulouse.

254. This apparatus consisted of a rectangular metal chest, 0·41 met. long, 0·41 broad, and 0·82 deep, communicating by a pipe with a reservoir kept constantly full, and 9 met. above the chest; the different adjutages were fitted into one of the walls of the latter vessel. A rectangular opening was made, and pipes were so arranged round its sides, that charges of 0·20, 0·50, 1·0, 1·50, 2·00, and 3 met. above the experimental adjutage, could easily be procured. The adjutages were of two kinds; the diameter of one was 0·0155 met., and about 0·040 met. in length; and the diameter of the other was 0·020 met. and 0·050 met. in length. For the purpose of determining very exactly the velocity of issue from the adjutage, M. Castel set up a horizontal plate or floor, in the centre of which was a groove 0·10 met. wide, into which

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1 See Recherches Expérimentales sur la Communication Latérale du Mouvement dans les Fluides, 1797. 2 Anciens Mémoires de l'Académie de Toulouse, tom. ii., 1784. Experiments on the Motion of Fluids.

The jet passed; its range was measured by means of a graduated rule fixed to, and very near the plate. This power was the ordinate of the path described by the jet; 1:140 met. was the abscissa; and from these two co-ordinates the velocity of projection was deduced by 177.

255. The same adjustment, under charges which varied from 0:21 to 3:03 met., gave wastes proportional to $\sqrt{H}$, and consequently coefficients which were sensibly the same. A slight increase will be observed under the charge of 3 met.:—

| Adjustment of 0:0155 met. | Adjustment of 0:020 met. | |--------------------------|-------------------------| | **Charge** | **Coefficient of Waste** | **Velocity** | | Met. | | | | 0:215 | 0:946 | 0:963 | | 0:483 | 0:946 | 0:965 | | 0:992 | 0:946 | 0:963 | | 1:492 | 0:947 | 0:965 | | 2:006 | 0:946 | 0:956 | | 3:030 | 0:947 | ... |

The higher charges are given in the tables for each series of diameters which have furnished the maximum waste.

With respect to the coefficients of the velocity, they would have also been sensibly the same had it not been for the resistance of the air. As this resistance diminishes the throw or range of the jet, and increases as the charge increases, we should expect in the calculated coefficients a diminution varying with the charge, although in reality there was no actual diminution in the velocity with which the fluid issued or tended to issue. In the following table we compare the coefficients both of the waste and of the velocity, obtained by the same series of adjustments, which differed only in their angle of convergence. Five or six different coefficients obtained from five or six different charges, very nearly the same, have been taken so as to determine a mean coefficient:

| Angle of Convergence | Coefficient of Waste | Coefficient of Velocity | |----------------------|----------------------|------------------------| | Deg. min. | | | | 0 0 | 0:829 | 0:830 | | 1 36 | 0:866 | 0:866 | | 3 10 | 0:895 | 0:894 | | 4 10 | 0:912 | 0:910 | | 5 26 | 0:924 | 0:920 | | 7 52 | 0:929 | 0:931 | | 8 58 | 0:934 | 0:942 | | 10 20 | 0:938 | 0:950 | | 12 4 | 0:942 | 0:955 | | 13 24 | 0:946 | 0:962 | | 14 28 | 0:941 | 0:966 | | 16 36 | 0:948 | 0:971 | | 19 28 | 0:954 | 0:970 | | 21 0 | 0:958 | 0:971 | | 23 0 | 0:963 | 0:974 | | 29 58 | 0:966 | 0:975 | | 40 20 | 0:969 | 0:980 | | 48 50 | 0:974 | 0:984 |

256. From the facts here tabulated it follows,—1. That for one and the same orifice of issue, and under the same charge, beginning with 0:83 of the theoretical waste, the actual waste gradually increases in proportion as the angle of convergence increases, but only up to 13½°, where the coefficient is 0:95. Beyond this angle it diminishes, slowly at first, like all variables about the maximum; at 20° the coefficient is about 0:92 or 0:93. But after this the diminution becomes more and more rapid, and terminates as low as 0:55, which is the coefficient for an orifice in a thin plate, this last being the limiting position of converging adjustments, that, namely, in which the angle of convergence has attained its maximum value of 180°. Hence, then, the maximum waste will be obtained with a converging adjustment of from 13° to 14°.

257. The explanation of this is probably the following:—In conical adjustments the theoretical waste is altered by the attraction of the walls, which tends to augment, and the contraction which tends to diminish it, by diminishing the section of the vein a little below the issue. From the experiments of Venturi, it would appear that the fluid vein, at its entry into the adjustment, preserves its natural form of a conoid of 18° to 20°; so that the more the angle of adjustment approaches this value, and the nearer the walls are to the vein when, after experiencing its greatest contraction, it tends to dilate itself, and when it is wholly under the attractive action of the walls, the greater will be the waste, and the maximum of counter action will give the maximum of waste. But, again, at 10° of convergence, the exterior contraction begins to be sensible, and causes a diminution of the waste; it has already reduced it by five per cent. at 18°; and thus the angle of maximum waste should naturally be expected between these two values, or about 14°.

The adjustments of 0:020 met. diameter of issue, gave coefficients from $\frac{1}{10}$th to $\frac{1}{5}$th greater than the adjustments of 0:0155 met. An error of $\frac{1}{10}$th of a millimetre at least in estimating the diameter of the first would, in a great measure, account for this difference; and the difference very likely arose from an error of this kind. The diameters of the second adjustments, however, were very correctly measured.

2. In regard to the coefficients for the velocity, we find them increasing from the angle of 0°, like those of the discharge, up to the convergence of 10°; beyond that point they increase more rapidly; and when beyond the angle of maximum waste, whilst the waste diminishes the coefficients of velocity go on increasing and approach their limit 1. They are already nearly equal to 1 at 50°, and not far from it at 40°. Conical adjustments may, by their diverse convergence, form a progression, the first term of which is the cylindrical adjustment, and the last the orifice in a thin plate; the velocity of projection, increasing with the convergence, varies then from that of the tube additional up to that of a simple orifice, that is, from $0:82\sqrt{2gH}$ up to $\sqrt{2gH}$.

3. Comparing the coefficients of waste with those of velocity, or the successive values of $n$ and $n'$, and dividing the first by the second, we obtain the values for $m$, or the coefficients of the exterior contraction. As the angle increases from 0° up to 10°, the values of $m$ are sensibly = 1, and consequently there is no contraction; and notwithstanding the convergence of the walls, the fluid molecules issued parallel to the axis. But beyond 10° the contraction shows itself; it reduces more and more the section of the vein, and terminates by reducing it to an equality with that of the orifice in a thin plate, as is shown in the annexed table.

Experiment having shown that cylindrical adjustments produce their full and entire effect as respects waste, when their length is equal to 2½ times at least their diameter, M. Castel fixed the length of the conical adjustment at about 2½ times the diameter of issue; thus it was 0:040 met. for those of 0:0155, and 0:050 met. for those of 0:020, in order, as far as possible, not to complicate the results with the friction of the water against the walls. In order, however, to determine the effect of elongation, he projected two other series for the adjustments of 0:0155 met.; in one the common length was 0:03 met., and reckoned the minimum; in the other it was 0·10 met., a very useful practical dimension. These he did not complete. He had already, however, made some approximations to them; thus, for adjutages of 0·0153 met., he took five of the length of 0·035 met., and the result was 0·938 as the coefficient of waste; whereas with a length of 0·04 met., the coefficient was 0·936; another adjutage, in length 0·03 met., gave 0·941 instead of 0·938; and one of 0·024 met. in length gave 0·931 in place of 0·926; thus a diminution of the length has a slight tendency to augment the waste. But, on the other hand, with adjutages of 0·020 met. the waste increases with the length; thus, in lengths varying from 0·50 met. to 0·10 met., and angles under 11° 32', he had a coefficient of 0·965; under angles of 14° 12', one of 0·955; and under 16° 34', one of 0·950.

It thus appears that the effect of the lengths of adjutages is far from being ascertained, and new experiments require to be undertaken to determine its full extent.

(b) Flow through Conical Diverging Adjutages.

258. We have seen (in 172) that this adjutage was known to the ancient Romans, and that by it we obtain a larger flow in a given time than from any other pipe. Bernoulli studied the effects of these adjutages and subjected them to calculation, and in one of his experiments found that the real velocity at the entry of the adjutage was greater than the theoretical velocity in the ratio of 100 : 108; but it is to Venturi that we are principally indebted for the knowledge which we possess of the results which these adjutages give. The following is a table of results from Venturi's experiments:

| No. | Nature and Dimensions of the Tubes and Orifices | Time in which 4 Parts cubic feet were discharged | Parts cubic inches discharged in a minute | |-----|-------------------------------------------------|-----------------------------------------------|------------------------------------------| | 1 | A simple circular orifice in a thin plate, the diameter of the aperture being 1½ inches | Seconds | Inches | | | | | 41 | | | | | 10115 | | 2 | A cylindrical tube 1½ inches in diameter, and 4½ inches long | Seconds | Inches | | | | | 31 | | | | | 13378 | | 3 | A tube similar to B (fig. 55), which differs from the preceding only in having the contraction in the shape of the natural contracted vein | Seconds | Inches | | | | | 31 | | | | | 13378 | | 4 | The short conical adjutage A (fig. 55), being the first conical part of the preceding tube | Seconds | Inches | | | | | 42 | | | | | 9874 | | 5 | The tube D (fig. 55) being a cylindrical tube adapted to the small conical end A, the diameter being 3½ inches long | Seconds | Inches | | | | | 42½ | | | | | 9758 | | 6 | The same adjutage, the diameter being 1½ inches | Seconds | Inches | | | | | 45 | | | | | 9216 | | 7 | The same adjutage, the diameter being 2½ inches | Seconds | Inches | | | | | 48 | | | | | 8840 | | 8 | The tube consisting of the cylindrical tube of Exp. 2, placed over the conical part of A | Seconds | Inches | | | | | 32½ | | | | | 12760 | | 9 | The double conical pipe E, consisting of 1½ inches of 0·977 inches, and the length of the outer cone = 1·351 inches | Seconds | Inches | | | | | 27½ | | | | | 15081 | | 10 | The tube F, consisting of a cylindrical tube 3½ inches long, and 1·376 inches in diameter, interposed between the two conical parts of the preceding | Seconds | Inches | | | | | 28½ | | | | | 14516 |

259. It appears from these experiments of Venturi that when water is conveyed through a straight cylindrical pipe of an unlimited length, the discharge of water can be increased only by altering the form of the terminations of the pipe; that is, by making the end of the pipe A (fig. 53), of the same form as the venea contracta, and by forming the other extremity BC into a truncated cone, having its length BC about nine times the diameter of the cylindrical tube AB, and the aperture at C to that at B, as 18 to 10. By giving this form to the pipe, it will discharge more than twice as much water in a given time, the quantity discharged by the cylindrical pipe being to the quantity discharged by the pipe of the form ABC, as 10 to 2½.

260. M. Venturi also found that the quantities of water discharged out of a straight tube, a curved tube forming a quadrantial arc, and an elbowed tube with an angle of 90°, each branch having a horizontal position, are to one another nearly as the numbers 70, 50, 45. Hence we see the disadvantages of sinuosities and bendings in conduit pipes. In the construction of hydraulic machines, any variation in the internal diameter of the pipe ought to be carefully avoided, excepting those alterations at the extremities which we have recommended in the preceding paragraph.

261. In one of his experiments Venturi employed a mouthpiece identical to that in fig. 53, and not unlike

Venturi concludes, from his experiments, that the adjutage of maximum waste should have a length of nine times the diameter of the smaller base, and an angle of divergence or expansion of 5° 6'. Such an adapted adjutage would give a waste 2·4 times greater than an orifice in a thin plate, and 1·46 times greater than the theoretical waste. The dimensions also should vary with the charge. 262. Venturi's Lateral Communication of Motion in Fluids (249) may be explained by one of his own experiments. Let a pipe AC (fig. 54), about half an inch in diameter and a foot long, proceeding from the reservoir AB, and having its extremity bent into the form CD, be inserted into the vessel CDG, whose side DG gradually rises till it passes over the rim of the vessel. Fill this vessel with water, and pour the same fluid into the reservoir AB, till running down the pipe AC, it forms the stream EGH. In a short time the water in the vessel CDG will be carried off by the current EG, which communicates its motion to the adjacent fluid. In the same way, when a stream of water runs through air, it drags the air along with it, and produces wind. Hence we have the water-blowing machine, which conveys a blast to furnaces (art. 340), and which will be described in a future part of this article. The lateral communication of motion, whether the surrounding fluid be air or water, is well illustrated by the following beautiful experiments of Venturi's. In the side of the reservoir AB (fig. 54), insert the horizontal pipe P, about an inch and a half in diameter, and five inches long. At the point o of this pipe, about seven-tenths of an inch from the reservoir, fasten the bent glass tube oam, whose cavity communicates with that of the pipe, whilst its other extremity is immersed in coloured water contained in the small vessel F. When water is poured into the reservoir AB, having no connection with the pipe C, so that it may issue from the horizontal pipe, the red liquor will rise towards m in the incurvated tube oam. If the descending leg of this glass siphon be six inches and a half longer than the other, the red liquor will rise to the very top of the siphon, enter the pipe P, and running out with the other water, will in a short time leave the vessel F empty. Now, the cause of this phenomenon is evidently this: When the water begins to flow from the pipe P, it communicates with the air in the siphon oam, and drags a portion along with it. The air in the siphon is therefore rarefied, and this process of rarefaction is constantly going on as long as the water runs through the horizontal pipe. The equilibrium between the external air pressing upon the fluid in the vessel F, and that included in the siphon, being thus destroyed, the red liquor will rise in the siphon, till it communicates with the issuing fluid, and is dragged along with it through the orifice of the pipe P, till the vessel F is emptied (421).

263. Another beautiful experiment of Venturi, and similar to the above, was made in the following manner. He added three cylindrical tubes to the lower side of a conical diverging adjuitage like that in fig. 53, and which would give 0·137 cub. met. In 25°, the first pipe being at the contracted section, the second distant by one-third the length of the adjuitage, and the third pipe at a distance of two-thirds the same length. These tubes were dipped into a vessel of mercury, and while the flow was taking place, the mercury rose in the respective tubes to heights of 0·120, 0·046, and 0·0158 met.; if water had been in the vessel these heights would have been 1·63, 0·63, and 0·125 mets. respectively. From the theory of Bernoulli, the pressure in this experiment at the point of greatest contraction is expressed by —1·60 met.; observation gave it as —1·63 met.

264. Eytelewin also made several experiments on diverging adjuitages; as these bear directly on the practical application of the subject, we shall briefly mention them. He took a series of cylindrical pipes of different lengths, and 0·026 met. in diameter, which were successively fitted on to a vase full of water; at first singly, then carrying at its anterior extremity a mouth-piece like A in fig. 53, which had nearly the form of the contracted vein; afterwards, carrying at its other extremity an adjuitage B (in fig. 53), of the form recommended by Venturi; and lastly, furnished at once both with mouth-piece and adjuitage. The flow took place under a mean charge of 0·750 met., and the following table gives the principal results:

| Length of pipe | Coefficient of Waste of single pipe from | Waste of single pipe | |---------------|----------------------------------------|---------------------| | Metres | Experiment | Formula of pipes | | | | With mouth-piece | | | | With adjuitage | | 0·001 | 0·62 | 0·99 | | 0·026 | 0·62 | 0·97 | | 0·078 | 0·82 | 0·95 | | 0·314 | 0·77 | 0·86 | | 0·628 | 0·73 | 0·77 | | 0·942 | 0·68 | 0·70 | | 1·255 | 0·63 | 0·65 | | 1·569 | 0·60 | 0·61 |

The above table shows—1st, The ratio according to which the length of the pipes diminishes the waste; 2d, That the increase of the waste arising from the divergence which the mouth has at the entry of the pipes, diminishes in proportion as their lengths increase; 3d, The effect of divergence diminishes the waste, and that more rapidly as the pipes increase in length. Eytelewin took a pipe 6·28 met. long, and 0·26 in diameter, and found no difference in the waste, whether be employed or not a diverging adjuitage. When this adjuitage was fitted on to the reservoir, the waste was 1·18, the theoretical being 1. On adapting it to the mouth-piece, but without the intermediate tube, it rose to 1·55; the mouth-piece alone gave only 0·92. Thus, the effect of the adjuitage B (fig. 53), fitted on to the mouth-piece A of the same figure, augments the waste in the ratio of 0·92 to 1·55, or 1:1·69.

265. M. Venturi was induced to institute a set of experiments, in which he employed tubes of the various forms exhibited in fig. 55. The results of his researches are contained in the table of 258, for which we have computed the column containing the number of cubic inches discharged in 1 minute. The constant altitude of the water in the reservoir was 32·5 French inches, or 34·642 English inches. The quantity of water which flowed out of the vessel in the times contained in the first column was 4 French cubic feet, or 4·845 English cubic feet. The measures in the table are all English. 266. When a cylindrical tube is applied to an orifice, the oblique motion of the particles which enter it is diminished; the vertical velocity of the particles, therefore, is increased, and consequently the quantity of water discharged. M. Venturi maintains that the pressure of the atmosphere increases the discharge of water through a simple cylindrical tube, and that in conical tubes the pressure of the atmosphere increases the expenditure in the ratio of the exterior section of the tube to the section of the contracted vein, whatever be the position of the tube.

SECTION III.—ON THE FLOW OF WATER OVER WEIRS, OVERFALLS, OR DEPRESIONS.

267. If we make a rectangular opening with a horizontal base in the upper part of the side of a vessel, the top edge being removed, the water, when maintained at a constant level, will flow out of this opening in the form of a sheet, and constitute an over-fall or weir.

The flow of water over weirs is very important. To determine the nature of the results arising from the use of such an orifice, we have to look principally to foreign hydraulicians. Dubuat in 1779 made several experiments on overfalls 18 inches wide, and 6 inches deep; those of Poncelet and Lesbro (388), 36 in number, were very carefully performed at Metz; the head, or charge of water, varied from 3 of an inch up to 8 inches; and the width was constantly about 7 inches. Messrs Smeaton and Brindley conducted a set of experiments, made over a waste-board 6 inches wide, and from 1 to 6 inches deep. Dr Robson made several experiments on the same subject. In 1834 MM. Daubisson and Castel made a series of very accurate experiments, at Toulouse water-works, on overfalls discharging water from a rectangular canal 29 inches wide, and of a variable depth. The widths of the apertures ranged upwards to the full width, and the head varied from about 1 to 8 inches. Mr Ballard made a set on weirs, on the River Severn, at Powick, near Worcester, in September 1836. But the most recent experiments on weirs are a first series by Mr Thomas Evans Blackwell, performed on the Kennet and Avon Canal in July 1850, and a second series by Messrs Blackwell and Simpson, made at Chew Magna, Somerset, during the summer of the same year.

268. If we look attentively at a waste-board B (fig. 36), we shall find that the water at a short distance C above it assumes a curved form CD, so that its height immediately over the weir is not equal to AB but only to DB. Now, in the common theory the fluid particles are supposed to have the same velocity on their arrival at D, as if they had fallen freely down AD; and so on, all the particles in the vertical section, DB, are assumed to flow out with velocities due to their heights below A; so that as respects the velocity of exit, and the number of the fluid threads, and also as respects the discharge, the case of a weir on such a supposition would be identical with a rectangular orifice closed at D, and in which the water level extended without any curvature up to A. Call, then, Q the volume discharged or waste escaped in one second, l the breadth of the weir, H and h the charge on the lower and upper edge respectively, m the coefficient of reduction of the theoretic to the real waste, then we have (by Prop. IV., 184), \( Q = \frac{3}{8} \sqrt{\frac{2g}{l}} \cdot m \times (H\sqrt{H} - h\sqrt{h}) \).

269. However natural such a supposition might be, MM. Daubisson and Castel deduce from their experiments facts which show that the wastes are more exactly given by supposing that the escape took place as if from the whole height AB, the fluid being supposed without curvature up to A. On such a supposition, we have \( h = 0 \), and

\[ Q = \frac{3}{8} \sqrt{\frac{2g}{l}} \cdot mlH\sqrt{H} = 5\cdot35 \cdot mlH\sqrt{H} \] (Prop. III. 183).

In the articles on Weirs, \( g \) is given in feet.

Hence the escape over weirs is only a particular case of the flow by orifices in general, that, viz., where the charge on the upper edge is nothing. Both Bidone (404) and Poncelet (388) had already shown that this was the case, and that the coefficients which serve ordinary orifices can be adapted to weirs also, when the flow is made under analogous circumstances.

270. In the formulae given above, it has been supposed that the fluid above the sill was in repose, or rather above that point where the surface begins to be curved towards the sill. But this is not really the case, for before arriving at this point the water has received an initial velocity, which must be taken into account, as we have already done in Prop. XVI. 203. We must now add on to the head due to the velocity in the case where the water is in repose, and which is now only \( \frac{1}{2}H \) (Prop. V. 185), the head which would generate that velocity with which it arrives. Let this velocity be \( u \), and \( h \) the charge due to it; then, since \( h = \frac{u^2}{2g} = 0\cdot01552 \times u^2 \), we shall have the real velocity of issue

\[ \sqrt{2g(\frac{1}{2}H + 0\cdot01552 \times u^2)} = 5\cdot35 \sqrt{H + 0\cdot03494 \times u^2}; \]

\[ \therefore Q = 5\cdot35 \times ml \cdot H \cdot \sqrt{H + 0\cdot03494 \times u^2}. \] \( u \) denotes the mean velocity of the section of the water which approaches the weir; its exact determination is well nigh impossible; but as its value differs little from that of the velocity at the surface, the equality may be admitted, and will modify the value of the coefficient which is to be determined experimentally. Call, then, this new coefficient \( m' \), and \( w \) the velocity of the surface of the water; then,

\[ Q = 5\cdot35 \times mlH\sqrt{H + 0\cdot03494 \times w^2}. \]

271. In the following experiments M. Castel has put these formulae to the test. We must remember, however, that the expression for the waste has two variables—the width of the weir, and a function of the velocity or charge. Now, in order that the formula should have claim to be well-founded, it is necessary that the waste should be exactly proportional to each, in which case, the coefficient \( m' \) would be constant; if this should happen with the coefficient, it will be a test of the accuracy of the formula.

272. M. Castel in 1835 and 1836 made numerous experiments on this subject, and with extraordinary care, at the Water-works of Toulouse. The principal apparatus which he employed was a wooden rectangular canal or trough, 6 met. long, 0\textsuperscript{7}4 met. wide, and 0\textsuperscript{5}5 met. deep; at one extremity he received a supply of water, and the other was so arranged that he could fit on thin copper plates in which the weirs were cut. The breadth of the overfalls varied from 0\textsuperscript{0}01 to 0\textsuperscript{7}4 met.; the sill or base was always 0\textsuperscript{1}7 met. above the floor of the canal. The waste was received at pleasure and during a given time, into a zinc-plated cistern of 3\textsuperscript{2}0 cub. met. capacity; this vessel was the gauge basin; it was therefore graduated vertically with the greatest care. The time that... the water took to rise a certain height was noted by a chronometer indicating quarter seconds.

273. The charges or heights of water above the sill of the weir were gradually increased from 0'03 to 0'10 and even to 0'24 met., for narrow overfalls. The most important, and at the same time, the most difficult matter was to determine exactly the measure of the charges. In order to do this very correctly, M. Castel placed over the middle of the canal, and parallel to its length, a bar or rule EF (fig. 56), and which carried, at distances of 0'05 met., 10 vertical pointed brass rods a, b, c, d, e, f, g, h, i, k, graduated into millimetres, and capable of sliding up and down; on the edge of the guides was a vernier graduated into 10ths of a millimetre. Whenever he wished to make an experiment, the requisite amount of water was admitted into the canal, and the regime or regulation for the proper supply being duly attended to, he let down the system of 10 rods, and put their points as accurately as possible in contact with the curved surface of the water. On subtracting, then, the reading of each rod from the vertical distance between the horizontal bar EF and the sill P, he obtained the values of the ordinates of the curve which the fluid particles described, as they advanced to the centre of the overall B. These ordinates increased as they became more distant from B: at 0'2 or 0'3 or 0'4 met., the increase was sensible; the greatest ordinate was the true charge H, while the smallest was the charge immediately over the sill or $H-h$ = the thickness of the sheet of water over the base B. The mean results of these experiments are to be seen in the following table:

| Charge upon the Sill, in metres | Canal, 0'74 metres wide. | |-------------------------------|--------------------------| | | Coefficients, the Length of the Overfall being, in metres, | | 0'74 | 0'08 | | 0'24 | ... | | 0'22 | ... | | 0'20 | ... | | 0'18 | ... | | 0'16 | ... | | 0'14 | ... | | 0'12 | ... | | 0'10 | ... | | 0'08 | ... | | 0'06 | ... | | 0'05 | ... | | 0'04 | ... | | 0'03 | ... |

274. After having in great measure exhausted the observations on the above canal, M. Castel experimented on one 0'361 met. wide, narrowing up the former by two partition boards 2'24 met. long. At the entrance of this small canal, which was placed in the centre of the larger, there was formed under large discharges a minute fall, which would have introduced some slight modifications into the results obtained had the partitions been extended up to the extremity of the larger canal. M. Castel, both on this and the former canal, performed in all 494 experiments, each being repeated twice. In each case the values of Q, L, and H, being given immediately by experiment, the coefficient m was deduced from the formula $Q=5'35 \times mH \sqrt{H}$ (269). The mean values of the experiments on the second canal are set down in the following table. In this and the former table where blanks occur no observations were made:

| Charge upon the Sill, in metres | Canal, 0'361 metres wide. | |-------------------------------|--------------------------| | | Coefficients, the Length of the Overfall being, in metres, | | 0'36 | 0'29 | | 0'24 | ... | | 0'22 | ... | | 0'20 | ... | | 0'18 | ... | | 0'16 | ... | | 0'14 | ... | | 0'12 | ... | | 0'10 | ... | | 0'08 | ... | | 0'06 | ... | | 0'05 | ... | | 0'04 | ... | | 0'03 | ... |

275. After having tabulated these results, M. Castel applies the simple and common formula $Q=5'35 \times mH \sqrt{H}$ and first shows that the wastes Q are proportional to the function of the charge $H \sqrt{H}$. For this purpose he takes the twenty-two series of wastes obtained, each with the same width of weir, but under different charges; he reduces the discharges of each series to that which they would have been, if one of them, e.g., that under 0'08 met., had been taken for unity or 1; he reduces also the series of values of $H \sqrt{H}$, and places them side by side, as in columns 1, 2, 3 of the following table:

| Charges | Series of Wastes | Series of | |---------|------------------|-----------| | | | | | H | 1 | 2 | 3 | | Mat. | | | | 0-20 | ... | 3-96 | 3-98 | 3-95 | 4-01 | | 0-18 | ... | 3-38 | 3-39 | 3-38 | 3-42 | | 0-16 | ... | 2-83 | 2-84 | 2-83 | 2-87 | | 0-14 | ... | 2-31 | 2-32 | 2-31 | 2-34 | | 0-12 | ... | 1-84 | 1-84 | 1-84 | 1-86 | | 0-10 | ... | 1-40 | 1-40 | 1-40 | 1-41 | | 0-08 | ... | 1-00 | 1-00 | 1-00 | 1-00 | | 0-06 | ... | 0-65 | 0-65 | 0-65 | 0-64 | | 0-04 | ... | 0-354 | 0-354 | 0-354 | 0-345 |

The first two are given by the canal 0-74 met., with the overfalls 0-60 and 0-10 met. in length; the third is from a canal 0-36 wide, with an overall 0-05 met. long. The conclusions, then, derived from a comparison of the twenty-two series of wastes with each other, and with those derived from the function $H \sqrt{H}$ are—

1st. That, on exceeding the charge 0-06 or even 0-05 met., some higher charges being excepted, the difference between the numbers of the same horizontal line are very small; they rise only by a 100th or so; and thus, in a practical point of view, the difference may be taken as nothing; so that the relation between the wastes may be regarded as the same as that between the corresponding values of $H \sqrt{H}$.

2nd. That for charges of 0-05 met. and below it, the wastes decrease in a ratio less than $H \sqrt{H}$, and much less as the charge becomes more feeble, but only with the mean lengths; for when they are very small, or when they approach the width of the canal, the equality is again produced.

3rd. In some higher charges, especially with wide overfalls, the wastes increase in a less ratio than $H \sqrt{H}$. This result, which was scarcely perceptible in a canal of 0-74 met., became very prominent in one of 0-36 met., where the water with these charges and with these widths, arrived at the overfall with considerable velocity. Now, in these cases, and they always present themselves when the fluid section $f$ at the passage of the weir exceeds the fifth part of the section of the current in the canal, the wastes ought not to increase as $H \sqrt{H}$, but as $H \sqrt{H} + 0.03494 \times w^2$, so that the formula as given in 270 must now be used. Hence, in limiting the subject to overfalls, properly so called, that is, those in which the water has an initial velocity of arrival, $Q$ will be very nearly proportional to $H \sqrt{H}$.

276. The formula, however, is no longer so coincident with experiment when the widths of the overfall vary. Beginning with the width of the basin the wastes diminish with the width of the overfall, but more rapidly up to a certain point, beyond which they diminish, less rapidly. The following table will show this. The canal of 0-74 met. has 12 widths, which areas the numbers in the first column; the second column shows the progress in which the corresponding wastes diminish, which were obtained under charges varying from 0-06 to 0-10 met. In the case of the canal of 0-36 met., where ten widths were employed, those only are set down which were nearly analogous to those of the other canal. This series of relations indicates that in both canals the wastes follow one and the same law with respect to the widths of the overfalls; to the relative widths, however, of the respective canals, and not to the absolute widths.

277. Since for one and the same width of overfall, and on neglecting extreme cases, the wastes are proportional to $H \sqrt{H}$, the coefficients should be nearly equal, and tables of 273 and 274 show us that they are so. Strictly speaking, however, the coefficients of any vertical column, beginning with the higher charges, decrease, generally very slowly, down to a certain charge, beyond which they rapidly increase; this charge, which is about 0-10 met., will be a minimum. Further, the charges remaining the same, the wastes at first decrease more and afterwards less rapidly than the widths of the overfalls; it follows that under the same charge, beginning with the width of the channel, the coefficients go on diminishing up to a certain point, and then beyond this increase. At this point, then, there will also be a minimum, which takes place when the width of the overfall is nearly one-fourth that of the channel of supply. Thus, in the horizontal as well as in the vertical columns of 273 and 274, there is a minimum, and hence there is a common minimum. In its immediate neighbourhood the coefficients on each side of it are very little different from one another; the variation is small, and so may be regarded as constant. But beyond this distance, the differences become considerable; they exceed an 8th in the values; and hence the waste by overfalls cannot be properly given with a constant numerical coefficient in the expression $l H \sqrt{H}$, for, mathematically speaking, the expression would not be admissible. In practice, then, we would require the assistance of a very extended table of coefficients deduced from hundreds of experiments. But the nature of the motion which the coefficients follow, when properly looked at, will enable us to avoid this labour, and to deduce a few simple rules suitable to the different cases which present themselves.

278. From what has been said in 275, the formula $l H \sqrt{H}$ is not applicable, on the one hand, when the charges are under 0-06 met.; nor, on the other, when the product of the transverse area and the width of the overfall exceeds the fifth part of the section of water in the supplying canal. Between these limiting values the above expression may be very well employed when combined with a coefficient depending on the width of the overfall.

Reckoning from the width of the canal itself, the coefficients diminish with the width of the overfall, until it has reached the fourth part of that of the channel; they still increase although the widths continually diminish; and what is rather remarkable, the coefficients, when diminishing, follow the relative widths of the overfall with respect to that of the canal, whereas, when they are increasing, which occurs afterwards, they depend on the absolute widths.

Such being the case, there will be four cases to distinguish relative to the coefficient to be employed in actual practice.

1st. In the neighbourhood of the common minimum already mentioned, the coefficients vary very little from each other. From the experiments of M. Castel at Toulouse, from a width of overfall nearly equal to one-third of that of the canal, supposed to exceed 0-30 met., down to an absolute width of 0-05 met., the coefficients only varied from 0-59 to 0-61, the mean of which is 0-60. Therefore the variable coefficient in this case to multiply $l H \sqrt{H}$ will be $0.35 \times 0.60 = 3.21$. Hence, between the limits, we have $Q = 3.21 \times l H \sqrt{H}$, a formula which is well adapted for gauging small streams of water. 2d. When the width of the overfall is a maximum, that is, equal to the whole breadth of the canal, in which case it will be a weir or waste-board proper, the coefficients have a remarkable constancy. M. Castel, in his experiments with a barrier 0·17 met. high, found no difference between the coefficients obtained from charges varying from 0·03 to 0·08 met. With an overfall 0·225 met. high the coefficients varied only from 0·664 to 0·666, the charges varying from 0·031 to 0·074 met.; when he took the mean it was 0·665. Hence, since \(5\cdot35 \times 0\cdot665 = 3\cdot55775\), and calling L the width of the canal or length of the barrier, we have \(Q = 3\cdot558 \times LH^{\frac{1}{2}}H\), a formula which may be used advantageously in gauging large courses of water, and with charges varying from 0·04 to 0·08 met.; but for accuracy it is necessary that the charge be less than one-third the height of the barrier.

3d. For widths of overfalls comprised between that of the channel and the fourth part of the same, the coefficient of the expression \(5\cdot35 \times LH^{\frac{1}{2}}H\) will vary with the relative width, or the ratio of the width of the overfall to that of the canal. The annexed table of coefficients is the result. These coefficients have been found, not by direct experiment, but by taking proportional parts between the coefficients deduced from experiment, and those from the tables given in 273 and that of 274. The coefficients for both canals were separately determined by observation, in order to show that the same relative width has corresponding coefficients sensibly the same, although the real value of the width was in one canal nearly double that of the other, which was a clear proof, that above 0·25 met., or a fourth part of the width of the channel, the coefficients depend for their value on the relative, not on the absolute widths of the overfall.

4th. It is otherwise, however, when this width falls below a fourth that of the canal. Then, and when at the same time it is less than 0·08, or 0·06 met., the width of the canal has no further effect, and each absolute width of the overfall has its proper coefficient. Thus, on the canal of 0·36 met., as on that of 0·74, the widths of 0·05, 0·03, 0·02, 0·01 met., gave in both coefficients of 0·61, 0·63, 0·65, and 0·67 respectively.

279. With respect to the formula given in 268, or \(Q = 5\cdot35 \times mL(H^{\frac{1}{2}} - h^{\frac{1}{2}})\), where \(h = AD\) (fig. 56), the last column of the table in art. 275 lets us see that although the series of values tabulated for \((H^{\frac{1}{2}} - h^{\frac{1}{2}})\) differs little from those of the corresponding wastes, yet it follows them less closely than does the series of values for \(H^{\frac{1}{2}}H\). Hence, on this account, the second formula is less trustworthy than the first. Its application, besides, is made more difficult, for it has a term \(h^{\frac{1}{2}}\), the exact determination of which is very difficult.

280. In the formula, however, involving a term which is a function of the velocity with which the water flowing in the canal arrives at the overfall, this is scarcely the case. This formula is for the case of a high velocity, where the water makes its escape in virtue both of the charge H, and that due to a previously acquired velocity; it is expressed by \(Q = 5\cdot35 \times mLH^{\frac{1}{2}}H + 0\cdot03494 \times w^2\), which leads to 270.

M. Castel has deduced from his experiments values for the coefficient \(m'\). In these experiments he did not measure the velocity \(w\) of the surface current in the canal, but it may be determined from the mean velocity, which in this case is \(Q = eL(H + a)\), where L is in this case the width of the channel of supply, a the height of the sill above the bottom of the channel, and H the charge.

The velocity of surface water will be shown in 331 to be a 4th part higher than the mean velocity; hence we have, \(w = 1\cdot25 \times \frac{Q}{L(H + a)}\). With this value of \(w\), which is the highest that can be assumed, the coefficient \(m'\) differs only from \(m\) of the common formula when the velocity in the canal has a value sufficiently great that \(0\cdot03494 \times w^2\), which causes the difference between the two formulae, is comparable with H. Since \(0\cdot03494 \times w^2\) is very small, and under the square root, it will influence the quantity \(m'\) by little more than half its amount relative to H; if it be, for example, 2, 4, or 6 hundredths of H, the coefficients, everything else being the same, will only differ from each other by 1, 2, or 3 hundredths. In these three cases the section \(LH\) of the fluid sheet at the overfall is found to be respectively \(0\cdot1724, 0\cdot244,\) and 0·3 times the section of the supplying canal, or \(L(H + a)\); therefore, when the first of these two sections is less than the 5th part of the second, \(m\) and \(m'\) will be within \(1\cdot5\)th of each other. This was the case with two overfalls experimented on by M. Castel, where the width was less by one-half than that of the channel of supply. When it was greater, the term \(0\cdot03494 \times w^2\) had more influence, and the difference between the coefficients became greater. But although this term has been employed, it has not reduced to equality the coefficients \(m\) and \(m'\) for different widths of overfall; it has not reduced by one-half the differences which the values of \(m\) present, so that neither of the expressions \(Q = 5\cdot35 \times mLH^{\frac{1}{2}}H, Q = 5\cdot35 \times m'(H^{\frac{1}{2}}H + 0\cdot03494 \times w^2)\) can be employed with a constant coefficient, except where the width of the overfall is the same as that of the canal of supply.

In order then to determine this constant coefficient, M. Castel dammed up a canal of 0·74 met. by copper barriers, the height of which gradually decreased from 0·225 to 0·032 met.; and he found coefficients as noted in the following table:

| Height of barrier | Met. 0·08 | Met. 0·06 | Met. 0·05 | Met. 0·04 | |------------------|-----------|-----------|-----------|-----------| | 0·265 | 0·651 | 0·655 | 0·657 | 0·650 | | 0·170 | 0·640 | 0·647 | 0·650 | 0·654 | | 0·190 | 0·650 | 0·649 | 0·652 | 0·656 | | 0·083 | 0·635 | 0·642 | 0·646 | 0·650 | | 0·075 | 0·647 | 0·652 | 0·655 | 0·660 | | 0·041 | 0·667 | 0·664 | 0·665 | 0·668 | | 0·032 | 0·676 | 0·676 | 0·676 | 0·680 |

Those of the first five barriers are nearly the same, although otherwise they do not present the same regularity as ordinary overfalls: the mean term may be taken as 0·650. With respect to the last two barriers of 0·041 and 0·032 met., they are in a distinct class; they were very low, and the charges much exceeded their height; so that the case was as much one of a course of water flowing in an ordinary channel as when it passed over a waste-board. Since there was a near equality between the coefficients for one and the same barrier, the formula which was employed for their determination is in a manner confirmed. The experiments again on the canal of 0·36 with a barrier of 0·17 met. high, gave coefficients the mean of which was 0·654. Let the mean between those two means be taken, or 0·652, then since \(5\cdot35 \times 0\cdot652 = 3\cdot488\),

\[Q = 3\cdot488 \times LH^{\frac{1}{2}}H + 0\cdot03494 \times w^2,\]

where \(w\) is to be determined by direct observation. We have been very particular with these experiments of M. Castel, and Danhisson's remarks upon them, owing to the importance of the subject; for further information, see volume already mentioned. Sometimes channels are adapted to overfalls, in which case the water discharged will be confined, and be resisted by the friction of the bottom and sides; this retardation must react on the water and diminish the discharge. Into this case, however, we need not enter.

281. There is still a particular case of an overfall to be considered; it is that which Dubuat has termed a demi-deversoir or deversoir incomplet; we call such overfalls drowned weirs. A drowned weir, then, is when the tail of the water rises above the level of the waste board sill. Dubuat divided the height AC (fig. 57), the level of the canal above that of the sill C, into two parts, Ab and bC; the flow from the first he considered as from an ordinary overfall where Ab (=H) would be the charge, and the volume of escape by it would be $Q = 3.488 \times l \sqrt{H + 0.0349 \times w^2}$. In the second part he regarded the flow as taking place out of a rectangular orifice, the height of which was bC, and the charge equal to the difference of level Ab, between the surface of the upper and that of the lower course. The height of this lower surface above the sill is bC; and this height bC will be equal to $n - y$, representing n and y respectively by BD and CD. To the charge Ab or H must be added, as in the case of closed orifices, the height due to the velocity $u$ of the water in the canal, and then the velocity of issue will be $\sqrt{2g(H + 0.01553 \times w)}$, or $\sqrt{2g(H + 0.01 \times w)}$, where $w = 1.25 \times u$ (280). Hence for the volume discharged by this part we have

$$Q = 4.96 \times l(n-y) \sqrt{H + 0.01 \times w^2}.$$

Let the two partial discharges be added together so as to obtain the total waste Q;

$$\therefore Q = 3.488 \times l \sqrt{H + 0.0349 \times w^2} + 4.96 \times l(n-y) \sqrt{H + 0.01 \times w^2}.$$

282. Mr Smeaton made several experiments on weirs. He conducted his observations by noting the time in which a vessel of 20 cubic feet would be filled with water flowing over a notch 6 inches wide, of various depths, and with different charges, as given in the table.

The annexed table will show his results. We have thus a method of determining the coefficients. Thus, let the head, as in table 287, be 0.1042 feet, then from formula $Q = \frac{3}{2}lH\sqrt{2gH} = \frac{3}{2} \times 0.05 \times 0.1042 \times 8.024 \times \sqrt{0.1042} = 0.08995$ cubic feet per second; but the experiment gave 20 cubic feet in 326 seconds; hence, we have in one second 0.06135 cubic feet;

$$\therefore 0.08995 : 0.06135 = 1 : 0.682 = \text{coefficient}.$$

283. In September 1836, Mr S. Ballard made a series of experiments on the flow of water over weirs on the Severn at Powick, near Worcester. The experiments were made on a weir 2 feet long, which stood perpendicularly across a trough, and the results are exhibited in the following table:

| Depth of water flowing over weir, in inches | Cubic feet per minute, over 1 foot of weir | |--------------------------------------------|------------------------------------------| | 1 | 5.88 | | 11 | 7.14 | | 14 | 9.55 | | 12 | 12.37 | | 2 | 14.93 | | 21 | 18.29 | | 24 | 23.07 | | 27 | 27.00 | | 3 | 32.14 | | 31 | 34.81 | | 37 | 37.81 | | 32 | 41.47 |

At the commencement of these experiments the results could not be rendered satisfactory owing to the difficulty of observing the exact depth of water on the weir, the gauge on the side being rendered inaccurate by the action of capillarity. The method then adopted was to attach two needles, one a very little larger than the other, to the lower end of a nicely graduated gauge, so that on the water being admitted by a sluice, regulated by a screw, its level was adjusted until it just touched the longer needle, and occasionally by its uneven motions the shorter one, which thus gave the exact height of the water above the weir. The gauge tank was graduated vertically and capable of containing 300 cubic feet. It was thought that the perpendicular position of the board forming the weir might lessen the volume passing over it, and, to avoid this, a sloping board was substituted, inclining on the upper side from the top of the weir downwards. Under this new arrangement the volume discharged was increased, with 1 inch depth of water, from 5.88 cubic feet per minute to 6.76 cubic feet in the same time, or about 15 per cent. Experiments were afterwards made on a weir 1 foot long, and the volume discharged was found to be less than with the 2-feet weirs. This was attributed to the contracted stream caused by the direction of the course of water at the sides of the weir. Experiments were also tried with oblique and circular weirs, and the volume of discharge was found uniformly to increase with the length of the weir. See Civil Engineer and Architects' Journal, vol. xiv., p. 647.

284. The experiments on overfalls by Mr T. E. Blackwell, being conducted on a much larger scale, may, for practical purposes, be regarded as furnishing results even more trustworthy and valuable than those of Dubuat, MM. Danhisson, and Castel. The first set of experiments was made in July 1850, in a side pond of the Kennet and Avon Canal, the area of which measured 106,200 square feet. It was guarded by a lock at each end, so that no current disturbed the surface. The supply reservoir, which was distinct from the experimental one, did not furnish a supply equal to the volume that escaped. The deficiency was made up three or four times a-day from the upper lock. This, however, would make no appreciable difference in the result when the reservoir was so large and the time required so small. The subsidence of the water due to even the largest volume of 444 cubic feet discharged, would only be about 0.00418 feet over the whole surface, so that the mean fall would be no more than 0.00209 feet. At some small distance above the overfall, the depth of water was somewhat reduced by a submerged course of masonry which rose to within 18 or 20 inches of the surface; and the overfall was placed on the outer line of the dam, and not exactly in the line of one of the sides of the reservoir. From the measurements given of the head, taken at still water, and the corresponding depth of the sheet of water flow- Experiments on Overfalls, Kennet and Avon Canal.

| Overfall | Waste | Coefficients | |----------|-------|--------------| | Thin plate 2 ft. long. | 1765 | 4397 | 14277 | | Thin plate 10 ft. long. | 1765 | 4397 | 14277 | | Plank 2 in. wide, 3 ft. long. | 1765 | 4397 | 14277 | | Plank 2 in. thick, 6 ft. long. | 1765 | 4397 | 14277 | | Plank 2 in. wide, 10 ft. long. | 1765 | 4397 | 14277 | | Crest 3 ft. wide, 3 ft. long, slope 1 in. 12. | 1765 | 4397 | 14277 | | Crest 3 ft. wide, 3 ft. long, slope 1 in. 18. | 1765 | 4397 | 14277 | | Crest 10 ft. wide, 3 ft. long, slope 1 in. 18. | 1765 | 4397 | 14277 | | Crest 3 ft. wide, 3 ft. long, level, 6 ft. long. | 1765 | 4397 | 14277 | | Crest 3 ft. wide, 10 ft. long, level. | 1765 | 4397 | 14277 |

The results in the table of the first set of experiments represent the case of the discharge of water by an overfall from a large still reservoir. In the second set the discharges are analogous to the case of a weir in a river or in a running stream.

287. Mr Blackwell deduces from his experiments several important facts. In thin plates the coefficient is highest at the smallest head observed; it reaches the mean at a head of about 3 inches, but beyond this point it decreases as the head increases. With a plank 2 inches thick, and a head of 1 inch, the coefficient is less than the mean; the mean is earlier reached as the length of the weir becomes greater; the mean head is about 3 inches, the coefficient continues to rise higher than the mean till the head reaches about 9 inches, but after this it is depressed below it. He found that a head of about 4 inches gave a smaller outflow than could be obtained by interpolating the results with heads of 3 and 5 inches. It is difficult to explain the reason of the depression.

Mr Blackwell also made several experiments to determine the effect of converging wing walls, noting the results on a weir 10 feet long, with and without wings, the wings... making an angle of 54°. The mean coefficient for that without wings was .371, while that with wings was .459. The coefficients in the second table, up to a head of 3 inches, are below the mean, above that head they fluctuate considerably; but generally they keep above the main line.

These anomalies are difficult to be accounted for, since every care was taken in the experiments. They remain to be elucidated by further investigations.

The following table, compiled from various sources, gives at one view the results of different experimenters:

### Experiments on Overfalls

| Overfall 0-5 feet long. Smeaton and Brindley. | |-----------------------------------------------| | Heads | ... | ... | .083 | .1042 | .1146 | .1354 | .1927 | .2604 | .4166 | .4687 | .5417 | | Coefficients | ... | ... | .713 | .681 | .654 | .638 | .638 | .602 | .609 | .571 | .533 |

| Overfall 1-533 feet long. Dubuat. | |-----------------------------------| | Heads | ... | ... | ... | ... | .1482 | ... | .2666 | .3887 | ... | ... | .5627 | | Coefficients | ... | ... | ... | ... | .648 | ... | .624 | .627 | ... | ... | .630 |

| Overfall 0-650 feet long. Danhasson and Castel. | |------------------------------------------------| | Heads | ... | ... | .098 | .131 | .164 | .195 | .262 | .328 | .393 | .459 | .524 | .590 | | Coefficients | ... | ... | .632 | .624 | .620 | .617 | .616 | .617 | .620 | .624 | .628 | .633 |

| Overfall 0-6548 feet long. Poncelet and Lebros. | |-------------------------------------------------| | Heads | ... | ... | .033 | .066 | .099 | .1332 | .1998 | .2664 | .333 | .50 | .666 | .75 | | Coefficients | ... | ... | .626 | .625 | .618 | .611 | .601 | .595 | .592 | .590 | .585 | .577 |

| Overfall 3 feet long and 10 feet long. Blackwell. | |--------------------------------------------------| | Heads | ... | ... | .063 | .166 | .25 | .33 | .416 | .50 | .583 | .666 | .75 | | Coefficients | ... | ... | .742 | .738 | .636 | .635 | .625 | .592 | ... | .580 | .529 |

| Overfall 10 feet long. Simpson and Blackwell. | |----------------------------------------------| | Heads | ... | ... | .063 | .166 | .25 | .33 | .416 | .50 | .583 | .666 | .75 | | Coefficients | ... | ... | .608 | .682 | .725 | .745 | .780 | .749 | .772 | .602 | .781 |

288. According to Mr Beardmore, in his *Hydraulic and Tide Tables*, the best mode of gauging weirs is by means of a post with a smooth head, level with the edge of the waste-board or sill, and driven firmly in some part of the pond above the weir, which has still water. A common rule can then be used for ascertaining the depth, or a gauge, showing at sight the depth of water passing over, may be nailed on, with its zero at the level of the sill of the weir. Among practical engineers, gauging by a weir has been always justly held to afford the most certain and efficient result, especially for ascertaining the comparative discharges of streams. For correct gauging, the effects of wind and current must be destroyed, a thin-edged waste-board must be used, and a weir not so long in proportion to the width above it as to wire-draw the stream, else the water will arrive at the weir with an initial velocity due to a fall which is not estimated in the gauging, and the result will be most probably too small. A weir, for correct gauging, should always have a free fall over; but there are sometimes cases where measurements are required with drowned weirs—so called when the tail has risen above the level of the sill. In this case we have two conditions which have been already stated in 281. Mr Beardmore's *Hydraulic Tables* for overfalls proper, gives the discharge for one foot of length.

### SECTION IV.—EXPERIMENTS ON THE EXHAUSTION OF VESSELS

289. It is almost impossible to determine the exact time in which any vessel of water is completely emptied. When the surface of the fluid has descended within a few inches of the orifice, a kind of conoidal funnel, of which the air occupies the middle part, is formed immediately above the orifice. The pressure of the superincumbent column being therefore removed, the time of exhaustion is prolonged. The water falls in drops; and it is next to impossible to determine the moment when the vessel is empty. Instead, therefore, of endeavouring to ascertain the time in which vessels are completely exhausted, the Abbé Bossut has determined the times in which the upper surface of the fluid descends through a certain height: **HYDRODYNAMICS**

### Times in which Vessels are partly exhausted.

| Paris feet | Square feet | Inches | Feet. | Min. sec. | |------------|-------------|--------|-------|-----------| | 11·6668 | 9 | | | |

Comparison of the Results of Theory with those of Experiment.

| Diameter of the circular orifice | Depression of the upper surface of the fluid | Time of the depression of the surface by experiment | Time of the depression of the surface by the formula | Difference between the theory and the experiments | |----------------------------------|-----------------------------------------------|-----------------------------------------------------|-----------------------------------------------------|---------------------------------------------------| | Inches | Feet. | Min. sec. | Min. sec. | Seconds | | 1 | 4 | 7 29½ | 7 29¾ | 3¼ | | 2 | 4 | 1 59 | 5 59½ | 1¼ | | 1 | 9 | 20 21½ | 20 16 | 5 6 | | 2 | 9 | 5 6 | 5 4 | 2 60 |

It appears from this table also that the times of discharge, by experiment, differ very little from those deduced from the corrected formula; and that the latter always err in defect. When the orifices are in the sides of the reservoir, the altitudes \( H \) and \( h \) of the surface may be reckoned from the centre of gravity of the orifice, unless when it is very large.

290. In order to compare these experimental results with those deduced from theory, we must employ the formula of Prop. X. 196, where the expression for the surface descending through any space \( a \) in a given time is

\[ T = \frac{2A}{mS\sqrt{\frac{2g}{H-h}}} \]

Let there be, for example, a prismatic vessel, the horizontal section of which is a square of 0·975 met. to the side, the orifice in the bottom being 0·0271 met. in diameter; the depth of water in it over the centre of the orifice is 3·79 met.; in what time will the surface fall through a depth of 1·30 met., reckoning from the orifice?

Here \( A = (0·975)^2 = 0·9506 \) sq. met., \( S = (0·0135)^2 = 0·000577 \) sq. met., \( H = 3·79 \), \( h = 3·79 - 1·30 = 2·49 \) met., and \( m \) say 0·61;

\[ T = \frac{2 \times 0·9506}{0·61 \times 0·000577 \times \sqrt{\frac{2g}{2·49}}} = 450 = 7 \text{ min.} \]

Bosset, however, when operating with these data, found the time actually required to be 7° 25' 5", \( g \) being given in metres.

291. He also made three experiments with the same apparatus, the results of which, theoretical and practical, are noted in the following table:

| Diameter of orifice | Diminution of surface | Time of diminution from Experiment | Formula | |---------------------|-----------------------|-----------------------------------|---------| | Metres | Metres | Min. sec. | Min. sec. | | 0·0271 | 2·92 | 20 25 | 20 41 | | 0·0541 | 2·92 | 5 6 | 5 10 | | 0·0541 | 1·30 | 1 52 | 1 52 |

The calculated time is slightly in excess of the observed result, but this being small may be neglected; the difference probably arose from some error of observation. To exemplify the expression in 198 for the fluid surface falling down a certain depth, let the upper surface of a basin which is sensibly prismatic, have an area of 1000 sq. met.; the water issues by a sluice opening 0·65 met. broad by 0·085 met. deep. The vessel, when the sluice rises, is filled to a height of 2·70 met. above this opening; find the depth through which the surface will pass in one hour!

Since \( a \) of 198 is also \( \frac{tmS\sqrt{2g}}{A} (\sqrt{H-tmS\sqrt{2g}} - \frac{4}{A}) \)

\[ \text{Depth required} = \frac{3600 \times 0·70 \times 0·05525 \times 4·43}{4 \times 1000} = 0·9182 \text{ met.} \]

where the value of \( g \) is taken in metres, and \( m = 0·70 \).

Hence, volume escaped in the course of the hour = 1000 \( \times 0·9182 = 918·2 \) cub. met.

### SECTION IV.—EXPERIMENTS ON THE ESCAPE OF WATER PASSING FROM ONE RESERVOIR INTO ANOTHER.

292. When a reservoir contains water, an escape may take place by an orifice in the bottom or side, not only in the open air, but also into another vessel partly full of water. If the escape be considered under this latter point of view, the three cases which will occur have already been mentioned under Prop. XV. (204).

As an application of the second case of Prop. XV., we may take the canal of Languedoc. General Andreossy, in his history of this canal, gives in mean terms—

- Length of the lock from one gate to the other... 35·08 met. - Breadth of the lock (balging in the middle) from 6·50 met. to... 11·04 - Fall from the upper to the lower course... 2·274 - Horizontal section of the lock... 325·6 sq. met. - Section of one opening... 0·6286 - Height of the upper course above the centre of one opening... 1·949 met. - Depth of the same centre to the level of the lower course... 0·323 - Coefficient of contraction when two openings of the gate are open at the same time may be taken as (229)... 0·548

We may consider at first, the part of the lock which is below the centre of the orifice. The time of its filling, as determined by the rule of escapes in the open air (181), will, taking two openings, equal to

\[ \frac{325·6 \times 0·323}{0·548 \times 2 \times 0·6286 \times \sqrt{\frac{2g}{1·949}}} = 24·85. \]

For the part now which is above the centre of the orifice, we shall have, by the formula 204, \( g \) being in metres,

\[ \frac{2 \times 325·6 \times \sqrt{1·949}}{0·548 \times 2 \times 0·6286 \times \sqrt{2g}} = 298 = 4\text{ min. }58\text{ sec.} \]

Thus the time required to fill the entire lock will be the sum of these two times, or 323° = 5° 23'. But General Andreossy says that the time required to fill it was between 5° and 6°; the mean of which is 5° 30', which differs very little from the theory.

293. Experiments have also been made on a sluice-gate of the canal of Bromberg in Germany; Eytweil cites them in his Manuel de Mecanique et d'Hydraulique. The lock was 49·59 met. long, its breadth was from 6·29 met. to 9·10 met., and its section was 422 sq. met.; the height of one of the openings was 0·4185 met., and that of the other 0·5623 met., the breadth of each was 0·6277 met. The escape took place first by the one then by the other, so that there were two sets of experiments. In each the water was previously left to rise in the lock to 0·06 met. above the upper edge of the opening; then the edge of the first opening was below the level of the upper course by 2·223 met., and that of the second by 2·197 met. The time that the water took to raise itself a certain space, or till the vessel was full, was reckoned in seconds. The results are given in the following table:— The time of the partial elevations has been found by the formula \( t = \frac{2A}{mS\sqrt{g}} (\sqrt{H} - \sqrt{h}) \) (196), where \( m \) may be taken as 0.625. The value of \( H \) in each of these partial experiments is the sum of the elevations noted in the second column, and reckoned from the bottom of the column, comprising an elevation corresponding to the time indicated against it; \( h \) is the same sum, but without comprehending this elevation; thus, for the second of the experiments noted in the table, \( H = 0.3399 + 0.6277 + 0.6277 = 1.5953 \), and \( h = 0.3399 + 0.6277 = 0.9676 \). By comparing then the two last columns, the results of experiments are found to agree very closely with those of calculation. If the difference were large, it would only indicate the great difficulty in determining the exact time when the water ceased to rise in the lock, the elevation about the end of the time taking place by infinitely small degrees.

294. As an example of the third case of Prop. XV., we may consider the two contiguous locks in the double sluice-gate of a canal. When a boat which ascends a canal has entered into the lower lock by its lower gate, it is shut in; the flood-gates, however, separating it from the upper lock rise; the water therefore falls in the latter, but rises in the former, and continues to do so until a common level has been attained; the gate of separation is now opened, and the boat is introduced into the upper lock. Take for instance the double sluice-gate of Bayard, near Toulouse. Let the time be reckoned from the moment at which the water, on reaching the lower lock, has attained the centre of the opening of the flood-gate. \( H \) being = 4.14 m., \( h = 0.24 \) m., \( A = 205 \) sq. m., \( B = 215 \) sq. m., \( s = 1.249 \) sq. m. (for the two openings), and \( m = 0.548 \); then the time elapsing from the moment of the rising of the flood-gate till the water has attained the same level in both locks, will, \( g \) being taken in metres, be expressed by

\[ t = \frac{2 \times 205 \times 215 \times \sqrt{4.14 - 0.24}}{0.548 \times 1.249 \times \sqrt{2g} \times 420} = 137^{\circ} = 2m\ 17^{\circ}. \]

But observation shows us that this time is 2m 29^{\circ}. The difference of 12^{\circ} indicates that the flood-gates were not fully raised when the water reached the centre of their openings, and the formula supposes that this was the case.

295. The consideration of vessels divided into different compartments by partitions, or diaphragms pierced by orifices, present, during the escape of the fluid which they contain, divers phenomena which have given rise to interesting mathematical results; for an account of which we refer the reader to Daniel Bernouilli's Hydrodynamique, sect. viii., and to Bossut's work on the same.

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**SECTION V.—EXPERIMENTS ON VERTICAL AND OBLIQUE JETS.**

296. We have already seen that, according to theory, vertical jets should rise to the same altitude as that of the reservoirs from which they are supplied. It will appear, however, from the following experiments of Bossut, that jets do not rise exactly to this height. This arises from the friction at the orifice, the resistance of the air, and other causes which shall afterwards be explained.

**Altitudes to which Jets rise through Adjutages of different forms, the Altitude of the Reservoir being Eleven Feet, reckoning from the upper surface of the horizontal Tubes, mmP, opR.**

| Diameter of the horizontal tubes | Form of the orifice | Reference to Fig. 74 | Altitude of the jet when raised vertically, reckoning from m. | Altitude of the jet when inclined a little to the vertical | |---------------------------------|---------------------|----------------------|---------------------------------------------------------------|----------------------------------------------------------| | Inch. lines. | | | | | | 3 8 | Simple orifice | H | 2 | 10 0 10 | | 3 8 | G | 4 | 10 5 10 | 10 7 6 | | 3 8 | F | 8 | 10 6 6 | 10 8 0 | | 3 8 | Conical tube | E | 94 by 70 | 9 6 4 | | 3 8 | Cylindrical tube | D | 4 by 70 | 9 1 6 | | 0 94 | Simple orifice | M | 2 | 9 11 0 | | 0 94 | L | 4 | 9 7 10 | | | 0 94 | K | 8 | 7 10 0 | |

297. It appears, from the first three experiments of the preceding table, that *great jets rise higher than small ones*; and from the last three experiments, that *small jets rise higher than great ones when the horizontal tube is very narrow*. There is, therefore, a certain proportion between the diameter of the horizontal tube and that of the adjutage or orifice, which will give a maximum height to the jet. This proportion may be found in the following manner. Let \( D \) (fig. 58) be the diameter of the tube, \( d \) that of the adjutage, \( a \) the altitude \( B \) of the reservoir, \( b \) the velocity along the tube; and as the velocity at the adjutage is constant, it may be expressed by \( \sqrt{a} \). Now the velocity in the tube is to the velocity at the adjutage as the area of their respective sections, that is, as the square of the diameter of the one is to the square of the diameter of the other. Therefore, \( \sqrt{a} : b = D^2 : d^2 \), and, consequently, \( b = \frac{D^2 \sqrt{a}}{d^2} \). If there is another tube and another adjutage, the corresponding quantities may be taken as \( \Delta, \delta, a, \beta \), and we shall have the equation \( \beta = \frac{\delta^2 \sqrt{a}}{\Delta^2} \). If we wish, therefore, that the two jets be furnished in the same man-

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1. *Histoire du Canal du Midi*, par le Gen. Andreassy. ner, then if the velocity in the first tube leaves to the first jet all the height possible, the velocity in the second tube leaves also to the second jet all the height possible, and we shall have \( b = \beta \), or \( \frac{d^2 V}{D} = \frac{S}{A} \). Hence \( D^2 : A^2 = \)

\[ \frac{dV}{da} : \frac{S}{a} \]

that is, the squares of the diameters of the horizontal tubes ought to be to one another in the compound ratio of the squares of the diameters of the adjutages, and the square roots of the altitudes of the reservoir. Now, it appears from the experiments of Mariotte (Traité du Mouvement des Eaux), that when the altitude of the reservoir is 16 feet, and the diameter of the adjutage 6 lines, the diameter of the horizontal tube ought to be 283 lines. By taking this as a standard, therefore, the diameters of the horizontal tube may be easily found by the preceding rule, whatever be the altitude of the reservoir and the diameter of the adjutage.

It results from the last three experiments, that the jets rise to the smaller height when the adjutage is a cylindrical tube (see D, fig. 58), that a conical adjutage throws the fluid very much higher, and that when the adjutage is a simple orifice the jet rises highest of all.

298. From a comparison of the fifth and sixth columns of the table, it appears that a small inclination of the jet to a vertical line makes it rise higher than when it ascends exactly vertically. Bossut, for example, gave a very slight inclination to a jet which rose vertically to a height of 3-42 met., and found the height increased to 3-47 met.; but even then it still falls short of the height of the reservoir. When the water first escapes from the adjutage, it generally springs higher than the reservoir; but this effect is merely momentary, as the jet instantly subsides, and continues at the altitudes exhibited in the foregoing tables. The great size of the jet at its first formation, and its subsequent diminution, have been ascribed to the elasticity of the air which follows the water in its passage through the orifice; but it is obvious that this air, which moves along with the fluid, can never give it an impulsive force. In order to explain this phenomenon, let us suppose the adjutage to be stopped, the air which the water drags along with it will lodge itself at the extremity of the adjutage, so that there will be no water contiguous to the body which covers the orifice. As soon as the cover is removed from the adjutage, the imprisoned air escapes; the water immediately behind it rushes into the space which it leaves, and thus acquires in the tube a certain velocity which increases at the orifice in the ratio of the area of the section of the tube to the area of the section of the orifice. When the orifice is small in comparison with the tube, the velocity of the issuing fluid must be considerable, and will raise it higher than the reservoir. But as the jet is resisted by the air, and retarded by the descending fluid, its altitude diminishes, and the simple pressure of the fluid becomes the only permanent source of its velocity. The preceding phenomenon was first noticed by Torricelli.

299. The following table exhibits all that is necessary in the formation of jets. The first two columns are taken from Mariotte, and show the altitude of the reservoir requisite to producing a jet of a certain height. The third column contains, in Paris pints, 36 of which are equal to a cubic foot, the quantity of water discharged in a minute by an orifice 6 lines in diameter. The fourth column, computed from the hypothesis in art. 296, contains the diameters of the horizontal tubes for an adjutage 6 lines in diameter, relative to the altitudes in the second column. The thickness of the horizontal tubes will be determined in a subsequent section:

| Altitude of the jet | Altitude of the reservoir | Quantity of water discharged in a minute from an adjutage six lines in diameter | Parameters of the horizontal tubes suited to the two preceding columns | |---------------------|--------------------------|-------------------------------------------------|--------------------------------------------------| | Paris feet | Feet, inches | Paris pints | Lines | | 5 | 5 | 1 | 32 | | 10 | 10 | 4 | 45 | | 20 | 21 | 4 | 65 | | 30 | 33 | 0 | 81 | | 40 | 45 | 4 | 95 | | 50 | 56 | 4 | 108 | | 60 | 72 | 0 | 129 | | 70 | 86 | 4 | 131 | | 80 | 101 | 4 | 142 | | 90 | 117 | 0 | 152 | | 100 | 133 | 4 | 163 |

300. We have already seen that jets do not rise to the heights of their reservoirs, and have remarked that the difference between theory and experiment arises from the friction at the orifice, and the resistance of the air. The diminution of velocity produced by friction is very small, and the resistance of the air is a very inconsiderable source of retardation, unless when the jet rises to a great altitude. We must seek, therefore, for another cause of obstruction to the rising jet, which, when combined with these, may be adequate to the effect produced. Wolfius has very properly ascribed the diminution in the altitude of the jet to the gravity of the falling water. When the velocity of the foremost particles is completely spent, those immediately behind, by impinging against them, lose their velocity, and, in consequence of this constant struggle between the ascending and descending fluid, the jet continues at an altitude less than that of the reservoir. Hence we may discover the reason why an inclination of the jet increases its altitude; for the descending fluid falling a little to one side does not encounter the rising particles, and therefore permits them to reach a greater altitude than when their ascension is in a vertical line. Wolfius observes, in proof of his remark, that the diminution is occasioned also by the weight of the ascending fluid, that mercury rises to a less height than water; but this cannot be owing to the greater specific gravity of mercury; for though the weight of the mercurial particles is greater than that of water, yet the momentum with which they ascend is proportionally greater, and therefore the resistance which opposes their tendency downwards, has the same relation to their gravity, as the resistance in the case of water has to the weight of the aqueous particles.

301. The two following experiments on oblique jets are given by Bossut. When the height NS of the reservoir AB (fig. 59) was 9 feet, and the diameter of the adjutage at N, 6 lines, a vertical abscissa CN of 4 feet 3 inches and 7 lines, answered to a horizontal ordinate CT of 11 feet 3 inches and 3 lines. When the altitude NS of the reservoir

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1 This was also observed by Wolfius (Opera Mathematica, tom. I., p. 802, sched. iv). 2 De Motu Projectorum: Oper. Geometr., p. 152. 3 Traité du Mouvement des Eaux, part iv., disc. 1., p. 303. Experiments on the Motion of Fluids.

was 4 feet, the adjutage remaining the same, a vertical abscissa CN of 4 feet 3 inches and 7 lines, corresponded with a horizontal ordinate CT of 8 feet 2 inches and 8 lines. The real amplitudes, therefore, are less than those deduced from theory; and both are very nearly as the square roots of the altitudes of the reservoirs. Hence, to find the amplitude of a jet when the height of the reservoir is 10 feet, and the vertical abscissa the same, we have \( \sqrt{9} \text{ feet} : \sqrt{16} \text{ feet} = 11 \text{ feet 3 inches 3 lines} : 15 \text{ feet 4 lines}, \) the amplitude of the jet required. This rule, however, will apply only to small reservoirs; for when the jets enlarge, the curve which they describe cannot be determined by theory, and therefore the relation between the amplitudes and the heights of the reservoirs must be uncertain.

302. The following experiments on oblique jets were performed by MM. Michelotti and Venturi. When the height of the adjutage above a horizontal plane was 19-33 inches, the amplitude of projection, according to Michelotti, was 23-2 inches, with a simple orifice, and 20 inches with an additional tube. Venturi found that when the height of the water in the reservoir was 32\(\frac{1}{2}\) inches, and that of the adjutage above a horizontal plane 54 inches, the amplitude of projection was 81\(\frac{1}{2}\) inches with a simple orifice, and 69 inches with an additional tube.

303. The most recent experiments on the height and discharge of jets are those lately made by the Southwark Water Company in January 1844, and those by the Preston Water Company in March of the same year. One set of experiments of the first company was made in Union Street through stand pipes, hose, and jets—there being six stand pipes, each 360 feet apart, and connected—

| Stand pipes used | Length of hose | Diameter of jet | Height of jet | Volume per minute | |------------------|---------------|----------------|--------------|------------------| | 1 | 40 | 1 | 60 | 15-7 | | 2 | 40 | 1 | 45 | | | 3 | 40 | 1 | 40 | | | 4 | 40 | 1 | 35 | | | 5 | 40 | 1 | 30 | | | 6 | 40 | 1 | 25 | | | 1 | 80 | 1 | 40 | 16-0 | | 1 | 160 | 1 | 40 | 16-0 | | 1 | 40 | 1 | 40 | 42-1 |

The total distance, then, from the head at Battersea, where the pressure was 120 feet, was 16,500 feet, or a distance of nearly 3\(\frac{1}{2}\) miles.

The second set of experiments of the same company was made in Tooley Street, where the stand pipes were connected—

| Stand pipes used | Length of hose | Diameter of jet | Height of jet | Volume per minute | |------------------|---------------|----------------|--------------|------------------| | 1 | 9-inch main | 4,200 feet long.| | | | 2 | 15-inch | 3,000 | | | | 3 | 20-inch | 12,750 | | |

So that the total distance from the head at Battersea under the above pressure, was 19,950 feet, or nearly 4 miles.

Experiments on the Height and Discharge of Jets, made by the Southwark Water Company in January 1844.

In both experiments every service pipe, or other outlet, was kept shut. The stand pipes employed were 2\(\frac{1}{2}\) inches in diameter, and the length of the hose varied.

The pipes marked \(a\) are through 600 feet of 5-inch main, but a 4-inch main was fitted on close to the 5-inch main; those marked \(b\) are through 600 feet of 5-inch main, and 600 feet of 4-inch main, both in addition to the 19,950 feet of main already mentioned.

304. The experiments by the Preston Company were made, first, on a 6-inch main, under a pressure of 110 feet; and, second, on a 6-inch main, under a pressure of 46 feet.

The results for the first are—

| Height | Discharge in cubic feet | |--------|-------------------------| | With 1 jet \( \frac{1}{2} \)-inches, 57 feet, 12-5 per minute by day. | | 1 | 64 | | 2 jets | 56 | | 2 | 62 |

The results of the second are—

| Height | Discharge in cubic feet | |--------|-------------------------| | With 1 jet \( \frac{1}{2} \)-inch, 24 feet, 4-8 per minute by day. | | 1 | 28 | | 2 jets | 20 | | 2 | 25 |

305. At Leeds jets may be thrown from 60 to 70 feet high, and in the lower part of the town from 40 to 50 feet high, the pressure being 180 feet, and services in full draught. At New York the height of the water is 115 feet above high water, 105 feet above the lowest, and 60 feet above the highest streets; the distributing reservoir is distant 4 miles, and communicates with the city by a 36-inch main; yet the city fountains throw from 60 to 70 feet high. At Harlem, River Valley, a \( \frac{1}{2} \)-inch pipe and a \( \frac{1}{2} \)-inch jet, throw the water to a height of 110 feet. At Philadelphia the surface of water in the reservoirs is 98 feet above high water, 55 feet above the highest, and 93 feet above the lowest points of the city. The distance from the reservoir to the extreme point of mains and pipes is 6 miles, by a main from 20 to 22 inches in diameter. Yet the water will rise by night to a height of from 40 to 50 feet, while by day to a height of 25 feet.

SECTION VI.—EXPERIMENTS ON THE MOTION OF WATER IN CONDUIT PIPES.

306. It must be evident to every reader, that, when water is conducted from a reservoir by means of a long horizontal pipe, the velocity with which the water enters the pipe will be much greater than the velocity with which it issues from its farther extremity; and that, if the pipe has various flexures or bendings, the velocity with which the water leaves the pipe will be still farther diminished. The difference, therefore, between the initial velocity of the water, and the velocity with which it issues, will increase with the length of the pipe and the number of its flexures. By means of the theory, corrected by the preceding experiments, it is easy to determine with great accuracy the initial velocity of the water, or that with which it enters the pipe; but on the obstructions which the fluid experiences in its progress through the pipe, and on the causes of these obstructions, theory throws but a feeble light. The experiments of Bossart afford much instruction on this subject; and it is from them that we have arranged the following table, containing the quantities of water discharged by pipes of different lengths and diameters, compared with the quantities discharged from additional tubes:— 307. The third column of the preceding table contains the quantity of water discharged through an additional cylindrical tube 16 lines in diameter, or the quantity discharged from the reservoir into a conduit pipe of the same diameter; and the fourth column contains the quantity discharged by the conduit pipe. The fifth column, therefore, which contains the ratio between these quantities, will also contain the ratio between the velocity of the water at its entrance into the conduit pipe, which we shall afterwards call its initial velocity, and its velocity when it issues from the pipe, which shall be denominated its final velocity; for the velocities are as the quantities discharged, when the orifices are the same. The same may be said of the 6th, 7th, and 8th columns, with this difference only, that they apply to a cylindrical tube and a conduit pipe 24 lines in diameter.

308. By examining some of the experiments in the foregoing table, it will appear that the water sometimes loses parts of its initial velocity. The velocity thus lost is consumed by the friction of the water on the sides of the pipe, as the quantities discharged, and consequently the velocities, diminish when the length of the pipe is increased. In simple orifices, the friction is in the inverse ratio of their diameter; and it appears from the table, that the velocity of the water is more retarded in the pipe 16 lines in diameter, than in the other which has a diameter of 24 lines. But though the velocity decreases when the length of the tube is increased, it by no means decreases in a regular arithmetical progression, as some authors have maintained. This is obvious from the table, from which it appears, that the differences between the quantities discharged, which represent also the differences between the velocities, always decrease, whereas the differences would have been equal, had the velocities decreased in an arithmetical progression. The same truth is capable of a physical explanation. If every filament of the fluid rubbed against the sides of the conduit pipe, then, since in equal times they all experience the same degree of friction, the velocities must diminish in the direct ratio of the lengths of the tubes, and will form a regular arithmetical progression, of which the first term will be the final, and the last the initial velocity of the water. But it is only the lateral filaments that are exposed to friction. This retards their motion; and the adjacent filaments which do not touch the pipe, by the adhesion to those which do touch it, experience also a retardation, but in a less degree, and go on with the rest, each filament sustaining a diminution of velocity inversely proportional to its distance from the sides of the pipe. The lateral filaments alone, therefore, provided they always remain in contact with the sides of the pipe, will have their velocities diminished in arithmetical progression, while the velocities of the central filaments will not decrease in a much slower progression; consequently, the mean velocity of the fluid, or that to which the quantities discharged are proportional, will decrease less rapidly than the terms of an arithmetical progression.

309. Bossut made the following experiment to show, that, in open channels and pipes, even where there is a considerable slope, the flow very soon becomes uniform. He constructed a wooden canal 650 feet long, with a slope of 1 in 10, and having divided it into equal spaces of 108 feet each, found that the water traversed all the other spaces except the first in equal times.

310. But besides the resistance which the water experiences from the walls of the pipe, and which is the most considerable, there is also the resistance arising from contraction, as Eytelwein first showed. If the bores be large this resistance is insensible, but if small it is too appreciable to be neglected. Further, if the pipe have one or more bends in its length, a third kind of resistance will arise; the loss by this resistance will evidently be greater in the case of an angular than in that of a uniformly curved tube. Bossut, for example, took a straight pipe 0-027 met. in diameter, and 16-24 in length, and under a charge of 0-325 met., obtained 0-02084 cub. met.of a volume in one minute. He now coiled the same pipe six times round, and obtained a volume of 0-02040 cub. met. (Bossut, Hyd. § 659). Rennie (410) also, after the same manner, took a straight pipe half-an-inch in diameter, and 15 feet long, and having adapted it to a reservoir under a charge of one foot, obtained a volume of 1-9 cub. feet in 1 min. But having bent the same pipe so as to produce a series of concavities and convexities in a semicircle of a radius 3-25 in., and having fixed it to the same reservoir, he found a volume of 1-7 cub. feet in the same time. Thus the 14 bends diminished the waste in the ratio of 100 : 89; when the charge was quadrupled, the ratio was 100 : 88. Venturi long ago showed the bad effect of bends by a series of experiments on three tubes 0-38 met. in length and 0-033 in diameter; one was straight, the second was rounded at an angle of 90°, and the third was right-angled. The charge employed was one of 0-88 met., and a vessel of capacity 0-137 cub. met. was filled by them respectively, in 45°, 50°, and 70°. In illustration of the same fact, Rennie took a pipe 15 feet in length and one inch in diameter, and under a charge of 4 feet, found the following volumes discharged per minute:

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1 Phil. Trans. of Royal Society of London, 1831. 311. These retarding or resisting forces in the case of fluids, go by the name of friction, the word being taken from the consideration of solid bodies rubbing or sliding upon each other. As there are laws of friction for the latter, so there are also laws of friction or resistance for the former. In the case of fluids, then, passing through straight pipes, as deduced from experiments, the resistance is—1. Independent of the pressure; thus water moving in a pipe under a head of 100 feet, experiences as great a resistance as if the head were 50 feet. Dubuat showed this law from experiments on the oscillation of water in siphons. In connection with this, see, under River, the experiment in which two vessels are connected by a siphon. 2. The resistance is, at any one velocity, proportional to the surface exposed to the action of the flowing water. 3. The resistance varies directly as the square of the velocity nearly, the border being constant. The third law is more nearly expressed by adding the simple power of the velocity.

312. When a pipe is inclined to the horizon, as CDE (fig. 60), the water will move with a greater velocity than in the horizontal tube CDhf. In the former case, the relative gravity of the water, which is to its absolute gravity as Ff to Cf, or as the height of the inclined plane to its length, accelerates its motion along the tube. But this acceleration takes place only when the inclination is considerable; for if the angle which the direction of the pipe forms with the horizon were no more than one degree, the retardation of friction would completely counterbalance the acceleration of gravity. Thus when the pipe CF, 16 lines in diameter, was 177 feet, and was divided into three equal parts in the points D and E, so that CD was 59 feet, CE 118 feet; and when CP was to Ff as 2124 to 241, the quantity of water discharged at F was 5795 cubic inches in a minute, the quantity discharged at E was 5801 cubic inches in a minute, and the quantity at D 5808 cubic inches. The quantities discharged therefore, and consequently the velocities, decreased from C to F; whereas if there had been no friction, and no adhesion between the aqueous particles, the velocities would have increased along the line CF in the subduplicate ratio of the altitudes CB, DM, EN, and PO; AB being the surface of the water in the reservoir. The preceding numbers, representing the quantities discharged at F, E, and D decrease very slowly; consequently, by increasing the relative gravity of the water, that is, by inclining the tube more to the horizon, the effects of friction may be exactly counterbalanced. This happens when the angle fCP is about 6° 31', or when Ff is the eighth or ninth part of CP. The quantities discharged at C, D, E, and F will then be equal, and friction will have consumed the velocity arising from the relative gravity of the included water.

313. In order to determine the effects produced by flexures or sinuosities in conduit pipes, M. Bossut made the following experiments:

| Altitude of the Conduit Pipes—See Figures 61 and 62. | Quantities of Water Discharged in a Minute. | |---------------------------------------------------|------------------------------------------| | Feet. Inches. | Cubic Inches. | | 0 4 | 576 | | 0 4 | 1050 | | 0 4 | 540 | | 0 4 | 1030 | | 0 4 | 520 | | 1 0 | 1028 |

314. 1. The two first experiments of the preceding table show that the quantities of water discharged diminish as the altitude of the reservoir. This arises from an increase of velocity, which produces an increase of friction.

2. The four first experiments show, that a curvilinear pipe, in which the flexures lie horizontally, discharges less water than a rectilineal pipe of the same length. The friction being the same in both cases, this difference must arise from the impulse of the fluid against the angles of the tube; for if the tube formed an accurate curve, the curvature would not diminish the velocity of the water.

3. By comparing the 1st and 5th, and the 2d and 6th experiments, it appears that, when the flexures are vertical, the quantity discharged is diminished. This also arises from the imperfection of curvature.

4. It appears, from a comparison of the 3d and 5th with the 4th and 6th experiments, that when the flexures are vertical, the quantity discharged is less than when they are horizontal. In the former case, the motion of the fluid arises from the central impulsion of the water, retarded by its gravity in the ascending parts of the pipe, and accelerated in the descending parts; whereas the motion in the latter case arises wholly from the central impulsion of the fluid. To these points of difference the diminution of velocity may somehow or other be owing.

When a large pipe has a number of contrary flexures, the air sometimes mixes with the water, and occupies the highest parts of each flexure, as at B and C, fig. 62. By this means the velocity of the fluid is greatly retarded, and the quantities discharged much diminished. This ought to be prevented by placing small tubes at B and C, having a small valve at their top.

315. A set of valuable experiments were made by M. Couplet, and a detailed account of them given in the Memoirs of the Academy for 1732, in his paper entitled Des Recherches sur le Mouvement des Eaux dans les Tuyaux de conduite. These experiments are combined with those of the Abbé Bossut in the following table:— ### Results of the Experiments of Couplet and Bossut on Conduit Pipes, differing in Form, Length, Diameter, and in the Materials of which they are composed.—under different Altitudes of Water in the Reservoir.

| Altitude of the water in the reservoirs | Length or the conduit pipe | Diameter of the conduit pipe | NATURE, POSITION, AND FORM OF THE CONDUIT PIPES | |----------------------------------------|---------------------------|----------------------------|------------------------------------------------| | Ft. in. | Feet. | Lines. | | | 0 4 | 50 | 12 | Rectilineal and horizontal pipe of lead | | 1 0 | 50 | 12 | The same pipe similarly placed | | 0 4 | 60 | 12 | The same pipe with several horizontal flexures | | 1 0 | 50 | 12 | Same pipe | | 0 4 | 50 | 12 | The same pipe with several vertical flexures | | 1 0 | 180 | 16 | Rectilineal and horizontal pipe of white iron | | 2 0 | 180 | 16 | Same pipe | | 2 0 | 180 | 24 | Rectilineal and horizontal pipe of white iron | | 2 0 | 177 | 16 | Same pipe | | 13 4 | 118 | 16 | Rectilineal pipe of white iron, and inclined like the last | | 6 8 | 159 | 16 | Rectilineal pipe of white iron, and inclined like the last | | 0 9 | 1,782 | 48 | Conduit pipe almost entirely of iron, with several flexures both horizontal and vertical | | 1 9 | 1,782 | 48 | Same pipe | | 2 7 | 1,782 | 48 | Same pipe | | 0 3 | 1,710 | 72 | Conduit pipe almost entirely of iron, with several flexures both horizontal and vertical | | 0 5 | 1,710 | 72 | Same pipe | | 0 5 | 7,020 | 60 | Conduit pipe partly stone and partly lead, with several flexures both horizontal and vertical | | 0 11 | 7,020 | 60 | Same pipe | | 1 4 | 7,020 | 60 | Same pipe | | 1 9 | 7,020 | 60 | Same pipe | | 2 1 | 7,020 | 60 | Same pipe | | 12 1 | 3,600 | 144 | Conduit pipe of iron, with flexures both horizontal and vertical | | 12 1 | 3,600 | 216 | Conduit pipe of iron, with several flexures both horizontal and vertical | | 4 7 | 4,740 | 216 | Conduit pipe of iron, with several flexures both horizontal and vertical | | 20 3 | 14,040 | 144 | Conduit pipe of iron, with several flexures both horizontal and vertical |

316. In order to show the application of the preceding results, let us suppose that a spring, or a number of springs combined, furnishes 40,000 cubic inches of water in one minute; and that it is required to conduct it to a given place 4 feet below the level of the spring, and so situated that the length of the pipe must be 2400 feet. It appears from the table in art. 245, that the quantity of water furnished in a minute by a short cylindrical tube, when the altitude of the fluid in the reservoir is 4 feet, is 7070 cubic inches; and since the quantities furnished by two cylindrical pipes under the same altitude of water are as the squares of their diameters, we shall have by the following analogy the diameter of the tube necessary for discharging 40,000 cubic inches in a minute; \( \sqrt{7070} : \sqrt{40000} = 12 \) lines or 1 inch: 234 lines, the diameter required. But by comparing some of the experiments in the preceding table, it appears that, when the length of the pipe is nearly 2400 feet, it will admit only about one-eighth of the water, that is, about 5000 cubic inches. That the pipe, however, may transmit the whole 40,000 cubic inches, its diameter must be increased. The following analogy, therefore, will furnish us with this new diameter; \( \sqrt{5000} : \sqrt{40000} = 28.54 \) lines: 80.73 lines, or 6 inches 8½ lines, the diameter of the pipe which will discharge 40,000 cubic inches of water when its length is 2400 feet.

317. The following experiments on the quantities of water discharged by different pipes of various lengths, and with different adjutages, were made by M. Bossut at the public and private fountains of Mezières in October 1779:

318. Dubuat undertook a series of experiments to determine the laws of resistance of bends, and their intensity. In order to this he employed different pipes, first straight, and measured the charge necessary to obtain a certain vo- lume in a certain time; then bent into different forms, such that the central fillet would make a certain number of angles, and of a certain size; and anew verified the charge under which the same volume would be obtained from the bent pipes in the same time. The difference between the two charges for the same pipe, now straight, then bent, was evidently the charge due to the bends, and, consequently, the measure of their resistance. The experiments were 25 in number, the most important of which are set down in the following table:

| Pipes | Angles | Velocity of water | Resistance due to bends | Coefficients | |-------|--------|-------------------|-------------------------|-------------| | Diameter | Length | Number | Value | Degrees | Metres | Metres | Metres | Metres | | 0-0271 | 3'167 | 1 | 35 | 2'300 | 0-0293 | 0-0111 | | 0-0271 | 3'167 | 2 | 35 | 2'300 | 0-0405 | 0-0111 | | 0-0271 | 3'167 | 3 | 35 | 2'300 | 0-0574 | 0-0123 | | 0-0271 | 3'167 | 4 | 24'57 | 2'300 | 0-0406 | 0-0111 | | 0-0271 | 3'167 | 10 | 36 | 1'839 | 0-01598 | 0-0123 | | 0-0271 | 3'749 | 4 | 36 | 1'572 | 0-0444 | 0-0136 | | 0-0271 | 3'749 | 4 | 36 | 1'572 | 0-0444 | 0-0136 | | 0-0271 | 19'95 | 4 | 36 | 0'776 | 0-0108 | 0-0127 | | 0-0541 | 6'910 | 4 | 36 | 2'838 | 0-0783 | 0-0099 | | 0-0541 | 6'910 | 4 | 36 | 1'590 | 0-0360 | 0-0103 | | 0-0541 | 6'910 | (5) | 24'57 | 2'335 | 0-2339 | 0-0124 |

Dubuat concludes from these experiments that the resistance of bends is proportional to the square of the velocity of the fluid, to the number of bends, and to the square of the sine of the angle which the bends make with the straight line of direction. Mr Beardmore, in his Hydraulic and Tide Tables, has added, inversely as the hydraulic mean depth. Experimenters, however, are not agreed on this point (414).

SECTION VII.—EXPERIMENTS ON THE PRESSURE EXERTED UPON PIPES BY THE WATER WHICH FLOWS THROUGH THEM.

319. The pressure exerted upon the sides of conduit pipes by the included water has been already investigated theoretically in Prop. XX., Part II. The only way of ascertaining by experiment the magnitude of this lateral pressure, is to make an orifice in the side of the pipe, and find the quantity of water which it discharges in a given time. The lateral pressure is the force which impels the water through the orifice; and therefore the quantity discharged, or the effect produced, must be always proportional to that pressure as its producing cause, and may be employed to represent it. The following table is founded on the experiments of Bossut:

Quantities discharged by a Lateral Orifice, or the Pressures on the Sides of Pipes, according to Theory and Experiment.

| Altitude of the water in the reservoir. | Length of the conduit pipe. | Quantities of water discharged in 1 minute according to theory. | Quantities of water discharged in 1 minute according to experiment. | |--------------------------------------|----------------------------|---------------------------------------------------------------|---------------------------------------------------------------| | Feet. | Cubic inches. | Cubic inches. | Cubic inches. | Cubic inches. | | 1 | 30 | 176 | 176 | 171 | | 1 | 180 | 190 | 190 | 194 | | 2 | 20 | 244 | 244 | 240 | | 2 | 90 | 264 | 264 | 261 | | 2 | 180 | 269 | 269 | 266 |

It appears from the preceding table, that the real lateral pressure in conduit pipes differs very little from that which is computed from the formula; but in order that this accordance may take place, the orifice must be so perforated, that its circumference is exactly perpendicular to the direction of the water, otherwise a portion of the water discharged would be owing to the direct motion of the included fluid.

320. It is found by experiment that, for pipes under pressure, a velocity of 200 feet per minute is very good for working with, giving perhaps a better proportional discharge than if the pipe had a greater inclination, and consequently the water a greater speed. A velocity of 150 feet per minute will prevent depositions of mud, &c., in pipes and sewers.

321. At the West Middlesex Water-works, it is found that the friction of the pipes reduces the head of water between one-fourth and one-fifth. The Grand Junction Water Company's new engine, at Kew, works against a head of 205 feet, while the gauge on the other side of the stand, which indicates the back pressure from London, gives only 170 feet; showing a loss of head equal to 35 feet, by the draught on the great 45-inch main. At Philadelphia the friction of the pipes causes a diminution of 25 feet of head when the city is drawing water. We see from the above that in towns supplied with water there is a considerable loss of head.

322. In towns there cannot but occur a great number of bends in the supply pipes. These bends taking place, as they actually do in many cases, in a vertical plane, are subject to two disturbances in the discharge. The first is an accumulation of air at the top of the bends, and which can only be liberated by valves self-acting, or worked by hand at stated times, since much air enters the pipes with the particles of water. The second is, that the diameter of the pipe over its whole length should be duly proportioned, so that it may discharge the volume due to its diameter. Thus, at a distance \( x \), from the fountain head or reservoir \( y \), the nature of the locality may be such, that the pipe at \( x \) will be a very little above the situation \( y \), in which case the requisite volume of water will not pass out by the pipe \( x \), lower situated than \( x \), or that volume due to a fall from \( y \) to \( z \). Neither can the pipe be fully effective at \( x \), since it is not completely filled. It is clear then, that for the requisite volume to be discharged at \( x \), the pipe from \( y \) to \( x \) must be of larger dimensions than that from \( x \) to \( z \). The neglect of providing against these impediments has led to much inconvenience and serious disappointment. It is said that when Edinburgh was supplied with water from Comiston about 1740, MacLaurin, from calculation, promised eleven times, and Desaguliers six times, more than the inhabitants received; the latter gentleman, however, erected air vessels or chests at different distances along the main, and the city was well supplied. When the Edinburgh City Water-works were extended, about 30 years ago, and the main pipe made of iron, every precaution was taken by Mr Jardine, the engineer, to secure the proper supply. The first 18,300 feet of the main had a fall of 1 in 282, the diameter of the main being at the reservoir 20 inches, and at the end of the above distance 18 inches; the remaining distance of 27,900 feet had a fall of 1 in 97, and the diameter of the discharging main at the Castlehill is 15 inches. On the whole, the actual discharge is not greater than that which would be given by a pipe of a diameter equal to 15 inches, and of a uniform slope over the entire distance. Mr Jardine erected, at 14 different points, cast-iron vessels to receive the compressed air as it collected; by opening a cock at each point, every three or four days the air is liberated. This was one of the earliest works in iron on a great scale, and it is reckoned a model of its kind; the cost of the works at the present price of iron would only be Experiments on the Motion of Fluids.

323. Mr W. A. Provis, M. Inst. C. E., undertook several experiments in the year 1838, on the flow of water through small pipes. The pipes employed were leaden, 12 inches in diameter, drawn to a length of 15 feet, and soldered together with great care. The united pipe was stretched on a wooden beam, resting on upright posts, in such a manner that it could be depressed or kept horizontal over its whole length towards the discharging end. A stop-cock was inserted into the upper end of the pipe; the lower end fitted into a cistern 2 feet square in the bottom, and 3 feet high, graduated upwards from the centre of the bore of the cock. There was a second cistern above this last, which, by means of a pipe and cock, supplied the lower with water. To the lower end of the pipe was attached, by a universal joint, an open trunk or spout by means of which the water from the pipe could be turned at pleasure into or outside a receiver, the capacity of which was 4 cubic feet. By means of a valve at the bottom of the receiver the water could be discharged. When an experiment was to be made, the highest cistern at the upper end of the pipe was filled with water, and the graduated one below it was also filled to the proper height. An assistant kept the latter vessel at the constant required level, by a continued supply from the upper one; a second assistant turned the cock which discharged into the pipe at the given signal. The lower end of the spout was held by a third assistant, who turned the stream into the receiver when ordered. Before taking the result of an experiment, the water was allowed to run for some time, but after a uniform and steady motion had been acquired in the water, and the signal given, the discharge for the time was determined. It was found that considerable difference prevailed in the time required for the run of the water through the pipe, owing to the dryness or moisture of the inner surface, and also by the accumulation of air inside when the pipe was level or slightly inclined. When the pipe was level and its interior nearly dry, as was the case on commencing in the morning, the time required for the water to pass through the pipe was nearly 50 per cent. more than after the tube had been thoroughly wetted. In all the experiments where the pipe was level, or nearly so, and more especially in the longer lengths, there were frequent regurgitations of the water, owing to the accumulation of air within the pipe; the experiments, however, on the time in filling the receiver, were not begun till the stream had become regular and equable. In order to guard against error, duplicate results were taken, and if any discrepancy took place in the first two results, a third was added.

The mean results of these experiments are given in the following table, which shows us—1. That when the pipes were level, the volumes discharged were nearly inversely as the square roots of their lengths. 2. In comparing the discharge with that passed through the 100-feet pipe, the departure from this rule appears greatest in the case of the short pipes of 20 feet length, when under the pressure of the greatest head of water. With a head of 6 inches, the rule would give about 100th more than the observed discharge; with a head of 35 inches, the rule would give about a 9th more than the actual discharge. The intermediate heads show intermediate differences. 3. By giving the pipes a regular and uniform descent, the discharge is increased through the long pipes in a greater ratio than through the short ones. 4. By increasing the head of water pressing at the upper ends of the pipes, the increase of volume is nearly in the same ratio through the long and short lengths:

| Fall of Pipe | Lengths of Pipe | |--------------|----------------| | | 100 feet | 80 feet | 60 feet | 40 feet | 20 feet | | Level | | | | | | | 1 in 112 | 2-275 | 2-5 | 2-874 | 3-504 | 4-528 | | 2 in 112 | | | | | | | 3 in 112 | | | | | |

Head of Water = 35 inches.

(Volume discharged in one minute.)

| Level | | | | | | | 1 in 112 | 1-991 | 2-264 | 2-944 | 3-178 | 4-138 | | 2 in 112 | | | | | | | 3 in 112 | | | | | |

Head of Water = 30 inches.

| Level | | | | | | | 1 in 112 | 1-745 | 2-008 | 2-302 | 2-823 | 3-664 | | 2 in 112 | | | | | | | 3 in 112 | | | | | |

Head of Water = 24 inches.

| Level | | | | | | | 1 in 112 | 1-476 | 1-684 | 1-959 | 2-394 | 3-116 | | 2 in 112 | | | | | | | 3 in 112 | | | | | |

Head of Water = 18 inches.

| Level | | | | | | | 1 in 112 | 1-151 | 1-352 | 1-558 | 1-882 | 2-437 | | 2 in 112 | | | | | | | 3 in 112 | | | | | |

Head of Water = 12 inches.

| Level | | | | | | | 1 in 112 | .763 | .901 | 1-052 | 1-26 | 1-668 | | 2 in 112 | | | | | | | 3 in 112 | | | | | |

Head of Water = 6 inches.

| Level | | | | | | | 1 in 112 | | | | | | | 2 in 112 | | | | | | | 3 in 112 | | | | | |

324. As pipes are exposed to forces besides those arising from the included water, they must be made much stronger than the preceding experiments would seem to require. The thicknesses of iron and leaden pipes used in France in the time of Bossut, are given in the following table:

| Iron Pipes | Lead Pipes | |------------|------------| | Diameter | Thickness | Diameter | Thickness | | Inches | Lines | Inches | Lines | | 1 | 1 | 1 | 2 | | 2 | 3 | 3 | 3 | | 4 | 4 | 4 | 4 | | 6 | 5 | 5 | 5 | | 8 | 6 | 6 | 6 | | 10 | 7 | 7 | 7 | | 12 | 8 | 8 | 8 |

SECTION VIII.—EXPERIMENTS ON THE MOTION OF WATER IN CANALS.

325. By a canal is meant a regularly constructed water-course or channel, having a uniform bed with the same inclination at every part, and containing the same volume of water over its whole length. Let a horizontal plane or right line \( ab \), be drawn through any point \( a \) of the bed of a canal, and a vertical \( eb \), corresponding to any other point \( c \) of the bed, then the inclination in the length \( ae \) of the canal will be represented by \( eb = \text{length } ae \times \text{sine of inclination } bac \). Call, then, \( L \) the length of a portion of a regular canal, \( D \) the difference of levels between the two extremities of \( L \), then the sine of inclination of canal will always be expressed

\[ D = \frac{\text{length } ae}{L}. \]

Call, again, \( S \) the section of the canal perpendicular to the axis of the current; then if the canal be rectangular, \( l \) its breadth, and \( h \) its height, or depth of water, \( S = lh \); if the canal be trapezoidal, \( l \) the breadth of the bottom, and also \( a \) the slope of the lateral sides, \( S = (l + ah)h = (l + h \cdot \cos a)h \), where \( a \) is the inclination of the walls to the horizon.

Let also \( c \) be the perimeter of the section, then for a rectangular canal \( c = l + 2h \); if it be trapezoidal,

\[ c = l + 2h \sqrt{n^2 + 1} = l + \frac{2h}{\sin a}. \]

The ratio of \( S : c \), Dubuat calls the mean radius (341).

326. When water flows in a canal, gravity is a force continually acting upon its particles. If the water be in repose, the tendency of gravity to put the water in motion is destroyed, and the surface is horizontal. But when the surface is inclined, then motion takes place among the particles, and hence the hydraulic axiom that motion will take place among the fluid molecules of a course of water whenever the surface is inclined. The inclination will apparently be the immediate cause of movement, but it is the gravity of the particles which produces it.

327. This action of gravity on the particles will be illustrated in the following cases. Let both the bed and surface of the canal be parallel, then we shall have, as it were, the case of a body sliding down an inclined plane, the particles being not only that layer in immediate contact with the bed, but those also of all the upper layers. Hence, since in this case gravity cannot act with its full intensity, so much of its effect will be lost, and the particle will slide down the bed with an intensity expressed by \( g \sin f \), where \( g \) is the accelerating force of gravity, and \( f \) the inclination of the surface to the horizontal plane. This is the condition of all the molecules on such a canal. Let now the surface be no longer parallel to the bed, but let the former be more inclined than the latter, any fluid molecule whatsoever traversing a length parallel to the bed will be subject to the action of gravity, together with that due to the inequality of pressure at the two extremities of the length. But these two forces are together equal to \( g \sin f \). Let, again, the bed be horizontal, and the surface inclined, then the particles in contact with the bottom will have their action destroyed by the resistance of the bed, but every other particle not in contact with the bottom will have a moving force represented by \( g \sin f \), where \( f \) is the inclination of the surface to the horizon. Let, lastly, the bed slope upwards, then the particles covering the bed will be impelled upwards along the bottom, while all others will be urged downwards with a force equal to the difference between the pressures of two columns, one of which is the depth of the particle under the surface, and the other the distance from the surface of any point on the horizontal plane passing through the former particle. This difference will again be expressed by \( g \sin f \), which will be the moving force on the particle.

328. We see, then, that in all the above cases, each particle, on traversing a line of any layer, is urged by a force equal to \( g \sin f \), or this is the final velocity of a particle, or the accelerating force at the end of a unit of time as a second. It evidently varies as the inclination, for \( g \) is constant. Let \( A \) be \( \sin f \), then the accelerating force will be

\[ gA. \]

Now, were this accelerating force the only one acting on a fluid moving in a canal, it is evident that the motion or velocity would never become uniform; for, if so, then equal volumes would pass through each section in equal times. But experiment and observation show us, that after a certain time, the accelerating force is imperceptible, and the water flows on with a uniform motion. Bossut, (Hydrod. §§ 797, &c.), in order to prove this, took a wooden canal 200 met. long, with an inclination of 1 in 10, and divided the length into spaces of 33 met.; he found, that, with the exception of the first space, the water passed over all the others in the same time. It is evident, then, that some retarding force exists whereby the accelerating force is destroyed at every instant. Whenever, therefore, this accelerating force has been destroyed, the water will move onwards with the velocity which it had at the first moment of escape. The retarding force mentioned above may be regarded as arising from the resistance of the bed.

But besides this retarding force, there is another of considerable intensity; it is that arising from the resistance or friction of the sides, which is not to be neglected. Eytelwein, to show this, took a pipe 0'63 met. long, and at the end of 100 sec. it discharged 0'148 cub. met.; having now doubled its length, 1'26 met., the same volume was obtained in 117 sec. Hence, we have another hydraulic axiom,—when the mean velocity is uniform, the accelerating force is equal to the retarding forces.

329. We shall have an idea of the nature of these resistances by observing the motion of a stream of water; as soon as it enters its course the water spreads over, wets and clings to the surface of the bed, and the same is the case with the walls; it is over this watery coating that the whole fluid mass passes, each of its particles being affected by the first coating, and by each rubbing on another. Hence the mass takes a mean velocity. Dubuat, from a considerable number of experiments, showed that this resistance between the bed of a canal and the water passing through it, is altogether different from the friction of a solid body on another, for his results show that, in the case of fluids, the resistance of friction is independent of the pressure (311). His experiments seem to have been performed with great care and ingenuity; he tried the flow of water over glass, lead, pewter, iron, wood, and different kinds of earths, and found no variation in the friction. (Principes d’Hydraul. §§ 34 et 36.) This inference of Dubuat is generally set down, in books on hydraulics, as the first law of friction in fluids. Daubisson, however, questions the truth of the law, stating that the adherence of the water is equivalent to a single force, and which may be measured, like all other forces, by a weight. He cites also one of Dubuat’s own experiments, where a piece of white iron put in contact with a tranquil surface of water, required a force of between 4'70 and 5'07 kil. on the square metre, to withdraw it from the water. He also mentions as an objection to it, the fundamental experiment which served Venturi to establish his principle of the lateral communication of fluids mentioned in 262.

330. The movement of water in canals is similar to that of water in pipes, the only difference being, that the one has an exposed surface, while the other has not. The fact of canals having an exposed surface offers a third cause of retardation, for the surface layer of fluid particles rubs against the lower layer of atmospheric particles. Hence, the liquid threads are not all animated with the same velocity; those which are most distant from the walls will move quickest, and thus we should expect the particles on the surface in the middle of the canal to have the greatest velocity. This is not, however, the case; the particles moving with the greatest velocity are actually found to be a little under the surface, as we shall see, in 359, in the experiments to determine the velocity of rivers. The loss of this velocity arises from the friction of the fluid particles with those of the atmosphere. Hence, also, the particles in a perpendicular section of a canal have not the same velocity, but if a mean of all these different velocities be taken, we shall have the mean velocity of this section. Call \( v \) the mean velocity, and \( s \) the perpendicular section, then the volume discharged, or waste, in a unit of time, is \( Q = sv \).

331. It is a matter of great importance in a practical point of view to determine the ratio of the mean velocity to the velocity of the surface. A knowledge of this relation has always been considered of great value by hydraulicians, and many experiments have been undertaken for the purpose. Dubuat is to be trusted on this head. He made thirty-eight experiments on two wooden canals, 43 metres long; the form of one was rectangular, and 0·487 metres broad, the section of the other was a trapezium, the least base of which was 0·156 met., and the sides inclined at an angle of 36°20' to the horizon; the depth of the water varied only from 0·034 metres to 0·273 metres, and the velocity was from 0·16 to 1·30 metres. The deductions of Dubuat from these experiments are, that the relation of the velocity of the surface to that of the bottom is greatest when the mean velocity is least; that the ratio is wholly independent of the depth; the same velocity of surface always corresponds to the same velocity of the bed. He observed, also, that the mean velocity is a mean proportional between the velocity of the surface and that of the bottom. Call \( v \) the mean velocity, \( u \) the velocity of the bottom, and \( V \) that of the surface, then Dubuat's results may be represented by the equations, \( u = (\sqrt{V - 0·165})^2 \), and \( v = \frac{1}{2}(V + u) = (\sqrt{V - 0·082})^2 + 0·00677 \). M. Prony has discussed Dubuat's observations, and has adopted the formula \( v = \frac{V + 2·372}{V + 3·153} \). The corresponding values of \( V \) and \( v \) are set down in the annexed table. M. Prony takes a mean term, and for practice makes \( v = 0·85 \times V \), so that, for the mean velocity of a current, it will suffice to diminish the surface velocity by one-fifth. Hence, the mean velocity of water flowing in a current is equal to four-fifths the superficial velocity.

332. When the surface of a current flowing into a long and regular canal takes at last a constant inclination, which is the same as that of the bed, in which case they will both be parallel, then all the transverse sections will be equal, and the mean velocity will be the same in each. The water moving in a canal under such circumstances is said to have a uniform motion. But when the inclination of the surface varies from one point to another, the surface is no longer parallel to the bottom, and so at divers points of the canal the transverse sections, and, consequently, the velocities also will vary. Nevertheless, since the volume of water entering the canal, and taken at any point, remains the same, the section of the fluid mass will be constant, and the velocity will have at all times the same value. Everything, indeed, will be unchanged, and although we shall not have a uniform, there will at least be a permanent motion.

333. A canal, with the exception of canals employed in inland navigation, may receive supply at its head from a reservoir, basin, or portion of a river, the level of which is raised for this purpose by dikes or banks. When the head of the canal, at the point where it receives the supply, is open, it is called a canal with a free entrance; when it is furnished with a sluice, it is termed a sluiced canal.

334. In the case of a canal with a free entrance, the water on entering forms a fall; its level lowers, then gradually rises by gentle undulations; beyond the influence of which it assumes a form nearly plane and parallel to the bed, the inclination being regarded as constant. The velocity accelerates during the lowering of the fall, retards while it is rising from the surface, and at last becomes sensibly uniform. Dubuat found that when the motion became uniformly regular, the velocity of the surface was nearly that due to the entire height of the fall, and that the height due to the mean velocity is equal to the difference between the height of the reservoir and that of the uniform section. (Principes d'Hydrod., § 177 et 178.) The fall which is formed at the entrance of a canal diminishes the depth of the uniform section, and lessens the volume of discharge. Hence, if a canal receive all the water that it is destined to convey, the fall must disappear.

Dubuat also infers from his experiments on this subject, that the velocity and the uniform section establish themselves at a certain distance from the reservoir, in the same manner as if uniformity commenced at the beginning of the canal. (Principes, § 177.)

335. The force of a current necessary to move a machine depends not only on the volume of water which it conveys, but also on the height from which it may fall. This force will be measured by the product of the volume and the height of the fall. The greater inclination that is given to a canal, and the more we augment the volume, a factor of the above product, the other factor, the fall, will be diminished for the same time; so that the product, increasing at first with the inclination, will diminish when the inclination is increased. There will thus be a maximum force, which it is very important to determine, and to put in practice. Thus, a volume of water equal to 11·86 cub. met. issues from a canal having a fall of 4·50 met., the effective fall being 4·06 met.; the product of these two quantities is 48·15, the corresponding inclination being 0·001045. But when the inclinations 0·0015, 0·002, 0·0025, and 0·003 met. are tried, the corresponding products will be 52·65, 54·68, 54·93, and 54·01. The fall of 0·003 met. has already given a diminution; and if the fall of 0·0026 met. be tried, the corresponding product is 54·88 met. Therefore, the maximum effect in this case will lie between the inclinations of 0·0025 and 0·0026 met.

336. When a canal (335), furnished with a sluice, receives water by the opening which is established at its head, it happens either that the upper edge of the opening is not completely and permanently covered by the water passing into the canal, or that it is so. If the charge over the centre of the orifice be greater than twice or thrice the height of the orifice, the upper edge will not be covered by the water, and the waste will be the same as if the canal did not exist. Bossut, for example, fitted on to a reservoir, having an orifice near the bottom, of dimensions 0·027 met. high, and 0·135 met. broad, a horizontal canal of the same sectional area, but 34 met. in length; the charges under which the escape took place were 3·80 met., 2·50 met., and 1·20 met., and he received at the extremity of the canal the same volume of water which it had taken in as when the canal was away altogether. (Hydrod., § 750.) The reason of this equality is, that when the water is forced along by means of a strong charge, and consequently with great velocity, the tendency of the vein to contract beyond the plane of the orifice renders the section smaller, and thus, on issuing from the orifice, it touches neither the sides nor the bottom of the canal; it is thus, as it were, projected into air, and the discharge remains the same as if this really took place. But beyond the contracted section, the fluid vein expands, and the resistance which it receives when coming in contact with the walls causes a diminution of flow. Where the contact of the walls takes place, however, the distance is too great from the orifice to react against the passing vein, and no reduction will arise in the waste. The dis-

---

1 See Prony's Traité des Eaux Courantes, 1802. charge will always be expressed by the formula already given in 183, \( m \sqrt{2gH} \), where \( l \) and \( h' \) are the breadth and height respectively of the opening; \( m \) having the same values as for orifices in a thin plate.

337. When, again, the charge is feeble, the opening is covered, and the water issuing from the sluice comes in contact with the walls of the canal, and suffers a retardation which it communicates to the fluid as it passes the opening. The waste, and therefore also its coefficient, are less in this than in the previous case. It sometimes happens, that when the charge is small, the sluice has no sensible effect, and the same volume may be given either when the sluice is wholly raised, or when it is plunged a little into the water. If, however, the depth at which it is plunged be considerable, and if the fluid vein on issuing be entirely covered by still water (as in fig. 49), the height due to the velocity will be the difference between the elevation of the two fluid surfaces. The depth at the sluice-gate is found by taking the height of the water in the canal when the movement has become very regular, the depth close to the sluice-gate being very small. If then \( h \) be the depth of water in the canal, \( H' \) the height at least of the sluice-gate above the sill of the entrance, the waste of the opening of the sluice-gate, and consequently, that of the canal, will be expressed by \( Q = m \sqrt{2g(H' - h)} \); but the expression for the discharge when the velocity has become sensibly uniform is

\[ S \left( \frac{\sqrt{2736 \frac{PS}{c}} - 0.033}{c} \right), \]

where \( S \) and \( c \) area s above, and \( p \) the sine of inclination;

\[ \therefore m \sqrt{2g(H' - h)} = S \left( \frac{2736 \frac{PS}{c}}{c} - 0.033 \right), \]

which is the general equation relative to canals furnished with a sluice-gate at their head.

338. The experiments of the Abbé Bossut were made on a rectangular canal 105 feet long, 5 inches broad at the bottom, and from 8 to 9 inches deep. The orifice which transmitted the water from the reservoir into the canal was rectangular, having its horizontal base constantly five inches, and its vertical height sometimes half an inch, and at other times an inch. The sides of this orifice were made of copper, and rising perpendicularly from the side of the reservoir, they formed two vertical planes parallel to each other. This projecting orifice was fitted into the canal, which was divided into 5 equal parts of 21 feet each, and also into 3 equal parts of 35, and the time was noted which the water took to reach these points of division. The arrival of the water at these points was indicated by the motion of a very small water-wheel placed at each, and impelled by the stream. When the canal was horizontal, the following results were obtained:

**Velocity of Water in a Rectangular Horizontal Canal 105 feet long, under different Altitudes of Fluid in the Reservoir.**

| Altitude of the water in the reservoir | Ft. in. | Ft. in. | Ft. in. | Ft. in. | Ft. in. | Ft. in. | |---------------------------------------|---------|---------|---------|---------|---------|---------| | Vertical breadth of the orifice....... | ½ inch. | ½ inch. | ½ inch. | 1 inch. | 1 inch. | 1 inch. | | Time in which the number of feet in column seventh are run through the water... | Sec. | Sec. | Sec. | Sec. | Sec. | Sec. | | 5 | 3 | 3+ | 2 | 2+ | 3- | 21 | | 7 | 9 | 4 | 5 | 6+ | 42 | | | 10 | 13 | 7 | 9 | 11+ | 63 | | | 13 | 17+ | 11 | 14 | 18+ | 84 | | | 16 | 20 | 11 | 14 | 18+ | 84 | | | 23+ | 28+ | 16+ | 20 | 26 | 105 | |

339. It appears from column 1st, that the times successively employed to run through spaces of 21 feet each, are as the numbers 2, 3-, 5, 6, 7+, which form nearly an arithmetical progression, whose terms differ nearly by 1, so that by continuing the progression we may determine very nearly the time in which the fluid would run through any number of feet not contained in the 7th column. The same may be done with the other columns of the table.

If we compute theoretically the time which the water should employ in running through the whole length of the canal, or 105 feet, we shall find, that under the circumstances for each column of the preceding table the times, reckoning from the first column, are 6:350, 7:834, 11:330, 6:350, 7:834, 11:330 sec. It appears, therefore, by comparing these times with those found by experiment, that the velocity of the stream is very much retarded by friction, and that this retardation is less as the breadth of the orifice is increased; for since a greater quantity of water issues in this case from the reservoir, it has more power to overcome the obstacles which obstruct its progress. The signs + and - affixed to the numbers in the preceding table, indicate that these numbers are a little too great or too small.

340. It is of the utmost importance to determine the form of canal which will convey the greatest possible volume of water under a given area of transverse section. In order to this it must be remembered, that the volume conveyed along the canal is greatest when the transverse section of the fluid mass is greatest, and the wetted perimeter least. A figure, therefore, must be selected, which, under the same perimeter, presents the greatest surface. By geometry, we know the regular polygons fall under this class of figures; and as a circle is nothing else than a polygon with an infinite number of sides, circular and semi-circular channels are also included in it. But when a semi-polygonal figure is chosen for the channel, the utility diminishes as the sides of the polygon diminish. The most easily executed are the regular semi-hexagon, the semi-pentagon, and the semi-square.

341. The question of figure then for canals which will convey the greatest volume of water, is reduced to taking, among all the trapezia with sides of a determinate slope, that one which has the greatest section for a given wetted perimeter; or (which is the same thing), that which has the greatest hydraulic mean depth. We shall find that this section is a semi-hexagon. The transverse area, therefore, or \( S = (l + m)h \) (see 325). This expression is to be a maximum, and, consequently, its differential will be zero;

\[ \frac{dS}{dh} = l + h \frac{dl}{dh} + 2mh = 0; \]

or, \( k \cdot dl + l \cdot dh + 2nh \cdot dh = 0 \).

But (325) \( e = l + 2h \sqrt{n^2 + 1} \) is to be differentiated, and \( e \) is a constant;

\[ \frac{de}{dh} = 0 = \frac{dl}{dh} + 2\sqrt{n^2 + 1}; \text{ or, } 0 = dl + 2 \cdot dh \cdot \sqrt{n^2 + 1}; \]

and on substituting the value of \( dl \) from the former in the latter equation, we have \( I = 2h(\sqrt{n^2 + 1} - n) \); giving \( I \) also its value in the expression for the transverse section, we have \( S = h^2(2\sqrt{n^2 + 1} - n) \); so also \( c = 4h\sqrt{n^2 + 1} - 2nh \).

\[ \frac{S}{c} = \frac{h^2(2\sqrt{n^2 + 1} - n)}{2h} = \frac{h}{2} \]

hydraulic mean depth (325); therefore, in all trapezoidal channels of the best form with certain given slopes and area, the hydraulic mean depth is one-half the depth of the water.

The transverse section of the maximum discharging channel may be thus obtained. Suppose that ABCD (fig. 63) is the regular channel required, draw EB, EC, dividing the section into triangles, of which AEB, EDC are equal in every respect; and let EP, ER, EQ be perpendiculars upon AB, DC, and BC from E. Then we have

\[ \frac{S}{c} = \frac{h}{2} \text{ or } S = c \cdot \frac{h}{2} \]

that is,

\[ = AB \cdot \frac{EP}{2} + DC \cdot \frac{ER}{2} + BC \cdot \frac{EQ}{2} = (AB + BC + CD) \cdot \frac{h}{2} \]

or \((AB + BC + CD) \cdot \frac{EP}{2} = (AB + BC + CD) \cdot \frac{h}{2}\);

\[ \therefore \frac{EP}{2} = \frac{h}{2} \text{ or, } h = EP. \]

A circle, then, with E as centre, and radius equal to EP, EQ, or ER, will pass through P, Q, and R. Generally, therefore, to find the maximum discharging section, describe a circle with any radius, draw a tangent to serve as the base of the section parallel to the horizontal diameter, and from the extreme points of the base draw two tangents of the requisite inclination till they meet the horizontal line; the figure comprised by the three tangents will be the transverse section required. The dotted tangents \( mo, rp \), and the base \( po \), mark another section for the same thing, but with the sides sloping at a different angle. Mr Neville gives this construction in his *Hydraulic Tables*, p. 129.

342. Canals for inland navigation, aqueducts, and wooden canals or courses for conveying water to manufactories, are invariably made of a rectangular form, owing to the simplicity, facility, and economy of their construction; but the semi-hexagonal section is that which conveys the greatest volume of water.

343. M. Genieys, in his *Essais sur les Moyens de Conduire, d'Élever, et de Distribuer les Eaux*, pays particular attention to the inclination of canals and aqueducts. He endeavours to find that velocity which best suits the nature of the soil, but specially one that would maintain the salubrity of the water. He gives 35 centimetres, or 13·75 inches per second, for the minimum velocity. M. Girard, in his *Rapport sur le projet général du Canal de l'Ouercq*, adopts a less velocity; he proposed at first to regulate the inclination of the Canal de l'Ouercq according to the law represented by the co-ordinates of the funicular polygon (see *Mechanics*); but the inclinations per metre for the upper (00000625 met.) and lower (00001236 met.) part of the canal were found insufficient. Dubuat thought that the smallest inclination capable of maintaining the mobility of water is 1 in 1,000,000; the motion is scarcely perceptible at 1 in 500,000. While experimenting on an artificial canal set to 1 in 9288, he found the mean velocity to be nearly 6 inches per second, and 7 inches per second in a drain near Condé inclined at 1 in 27,000.

The most recent observations on the ancient aqueducts show that the inclination varied from 1 in 432 to 1 in 648. The fall of the new river or canal which conveys water to London is only 0·25 feet per mile, or 1 in 21,120; its motion, about one-half mile per hour, is too small, since during summer the temperature of the water is raised.

344. The practical application of a fluid such as water flowing through orifices in the sides or bottoms of vessels, in pipes, and open canals, or over weirs, is of the utmost importance. Besides those instances which have already been mentioned for the purpose of illustration, we may state the following. An example, on a large scale, of water issuing through an orifice is to be found in the sluices constructed in tidal harbours for sweeping away at low water the deposit that generally accumulates there. Were it not for some effectual remedy of this kind, many of the most important tidal harbours on the English coast would have been long ere now completely silted up. Smeaton introduced these sluices into Britain from the low countries. He threw an embankment across some part which was wholly covered at high water, so that at the ebbing of the tide a considerable volume of water might be caught in a reservoir. At low water the sluices in the bank were drawn up, and the water allowed to escape among the mud. But this method soon fell into disrepute, as the area of the backwater itself soon became silted up. Smeaton, however, improved upon his original plan, by erecting a second bank altogether or nearly perpendicular to the first, by which the back-water was divided into two areas, each of which was made to cleanse the other, while both kept the harbour clear. The harbours of Ramsgate, Dover, and Hartlepool, are familiar illustrations of this method. (See Smeaton's *Reports*, vol. ii., pp. 202-209; and Sir John Rennie *On Harbours*.)

Other examples are seen in the filling of canal locks, and in the supplying of manufactories and other works from city pipes. In the latter case, a very small orifice is generally perforated in a disc, which is closed up so as to secure it from any unfair interference.

But perhaps one of the best instances of water flowing through orifices is to be found in districts watered by means of canals. Many examples of this kind are to be found in Captain Baird Smith's work on *Italian Irrigation*. In the irrigated districts of Italy it was long a matter of importance to determine the unit of volume of water, and some means of regulating the due supply to each proprietor; so that, on the one hand, neither the government nor landlords might be defrauded, nor, on the other, occupants of lands suffer any injustice. The earlier units were at first only orifices of a fixed area in the side of the canal, without any reference to the head or charge under which the issue took place; but the variations of the water-level in the canal, afterwards made it necessary that a constant pressure be maintained. The unit on the canal of Caluso was called *ruota* or wheel, and was defined as a square, the side of which is equivalent to 1'6702 feet: the upper edge of the outlet is locally said to be *a flor di acqua*, or level with the surface of the canal or reservoir, the discharge hence taking place under no pressure. The *modulo*, or volume per second discharged by the *ruota*, was 11'83 cubic feet. The volume measured by this *ruota*, which was introduced by F. D. Michelotti, was afterwards reduced to 10'84 cubic feet by his son Ignazio. Another of these units is the *oncia magistrale* of Milan; it is 0'655 feet high, 0'3426 feet broad, and has a constant pressure of 0'32944 feet over the upper edge of the outlet. The orifice is cut in a single stone slab, and that it may not be unfairly made use of, is girdled by an iron ring. The *modulo* or volume of the *onica magistrale*, as applied upon the Naviglio Grande, distributes 1851 cubic feet per second over a course of 31 miles. The honour of its discovery is due to Soldati of Milan, about the year 1571. It appears that for summer irrigation each cubic foot of water distributed per second suffices to water 61·8 acres, and that the annual rent for this volume is equivalent to L13·25. Great improvements have taken place in these outlets since they were first applied. One of the simplest and most beautiful is that founded on the effect of the double cistern. It is identical with fig. 49 of Prop. XV., the only difference being that in the vertical side of FE, an orifice in that side discharges in the open air.

Captain B. Smith gives us, in the case of the Grand Ganges Canal, an example on a very magnificent scale, of water flowing through an orifice. This canal, which was lately opened, has a course of 898 miles, is navigable throughout, and waters a district of country equal in area to 5,000,000 acres. The inclination of this canal is 1 in 3520; its depth is 10 feet, the breadth of its bed over the whole length is 140 feet, and it is intended to convey 6750 cubic feet per second. The slopes are not mentioned, but at from 1:5 to 1, calculation would give a volume of 7316 cubic feet. The inclination of this canal may, from its vast dimensions, be regarded as considerable, and such that the resisting velocity would wear down its sides, but it is supposed that this is prevented, and consequently the velocity diminished, by the rapid growth of aquatic plants on its bed, which is a serious impediment on all canal works.

M. Girard feared that the growth of plants would be an element in retarding the velocity of the water flowing in the Canal de l'Ourop, and adopted the inclination of 0.0000305 met., or 1 in 947.

As an example of water flowing in pipes, we need only mention the supplying of cities by mains and culverts or tunnels. The Croton Water-works supply the city of New York by a culvert, the section of which is not unlike the dotted fig. of 341; it is covered over its entire length, which is a means of preserving the water pure, and of a low temperature; its fall is 13½ inches, or 1:125 feet per mile; the waste is 60,000,000 gallons every 24 hours, or 111:1 cubic feet per second.

The projected supply from Loch Katrine for the city of Glasgow is to be conveyed by means of a culvert 8 feet in diameter, and inclined at 1 in 6336. The valleys which intervene are to be passed by a 48-inch cast-iron main; the fall over the whole length will be 1 in 1000, or 5:28 feet per mile.

It is a very important matter also to have, both for cities, manufactories, and for irrigating purposes, a constant variable discharge. The late Mr Thom, hydraulic engineer to the Gorbals Water-works, near Glasgow, devised an apparatus by which a constant discharge might be maintained through an orifice with a constantly varying charge over its centre.

The application of the subject to the case of weirs may be seen in the waste-boards thrown across rivers and streams. When weirs are seen in rivers, salmon-gaps are always made in the board, so that the fish may get up the stream. A very ingenious application of weirs is to be seen on the Manchester Water-works. The reservoirs are there supplied in part by the mountain streams, which, in rainy weather, carry down mud and filth, rendering the water unfit for domestic use. Hence the necessity of constructing weirs across several of these streams, to carry away the turbid water. In fine weather these supplying streams convey clear water, and discharge it by a weir at the top of a stone wall, which is a side of a reservoir open at the top. The opposite side of this reservoir is built at such a distance from the former that, in rainy weather, when the stream comes down with violence, its velocity urges it across the reservoir altogether, and all the dirty water is then conveyed to a compensation-pond, which supplies the mills of manufactories. Before and after its greatest velocity much muddy water would enter the reservoir, did not a man attend and put on a cover over the top.

SECTION IX.—FLOW OF WATER IN RIVERS.

It is only recently that the motions of rivers have been made the subject of scientific study; it may be said to date only from the year 1665, when the celebrated congress of philosophers met by appointment of the Roman and Florentine governments, to put an end to the quarrels which had taken place among the inhabitants bordering the Val de Chiana (the ancient Chiana Palus). The Chiana, lying between the Tiber and Arno had been alternately pushed backwards and forwards, till at last the district had become a noxious marsh. It was this same river which gave rise to the famous controversy in the Roman senate, when it was proposed to prevent the inundations of the Tiber, by throwing the Chiana into the Arno. What was once a marsh, however, is now one of the most fertile districts in Italy (10).

On the resistance of irregular forms and slopes of river beds, Dubuat made several experiments which are worthy of notice. He took different kinds of earths, sands, and stones, and placed them successively on the bottom of a wooden canal; on inclining the canal at different angles he varied the velocity of the water, and, therefore, also obtained a measure of the material set in motion.

For white clay the measure was 0:08 met.; fine sand 0:16; gravel of the Seine, size of a pea 0:19; gravel of the Seine, size of a bean 0:32; sea pebbles, an inch in diameter 0:65; flint stones, size of a hen's egg 1:00.

He afterwards covered the bottom of the canal with a layer of sand, and the sand moved with the water under a velocity of 0:3 met. After some time, the sand presented a series of beautiful undulations or ridges 0:12 met. broad. The grains of sand, forced on by the current, rose to the tops of the ridges, and after falling down by their own weight to the base of the next ridge, were again lifted to its summit. He found that half an hour was required for a particle to traverse a complete ridge, so that the rate of motion would be about 6 met. in twenty-four hours. It is in this way that the grains of sand move over the Dunes by the impulsion of the winds.

The effect of the velocity of the water in carrying away particles from the bed of a river will depend on their tenacity and size. With respect to the size, the volumes or weights of similar bodies decrease faster than their areas; and the pressure or force urging a body down a stream being proportional to the transverse section, is relatively greater the less is the volume; hence, the smaller the particles the less is the velocity required to move them. Mr Beardmore gives in his Hydraulic Tables the limit of bottom velocities in different materials.

When the beds of rivers, however, are protected by aquatic plants, they will bear higher velocities than those just now mentioned.

When the bed of a river is broad and deep, and the water moves slowly, the surface of the fluid will be nearly horizontal; but under a rapid current, in places where the river contracts, the surface assumes a very decided inclination. In a very decided slope of the surface to the horizon, the fluid surface becomes convex; it is concave when the inclination towards the horizon has diminished very much. Were the bed horizontal, and its outline constant, there would be an accelerated motion corresponding to the whole convexity of the surface; and a retarded one for the concavity. Were the bed uniformly inclined, there would only be an accelerated motion so long as the successive values of the inclination of the surface to the horizon were greater than that of the bed; otherwise, notwithstanding the convexity, the motion would be retarded.

356. The inequalities of the bottom have some influence on the form of the surface of the current. In general, the inequality at the surface will be smaller compared with that at the bottom, as the water is deeper and the velocity greater. Sometimes, also, in times of great inundations, when dikes about 8 feet high are covered, the surface is equal and the water passes over without any sensible depression.

357. The transverse section, then, of the surface of a river, presents the appearance generally of a convex curve, the summit thread being that which may be considered as having the greatest velocity. On each side of this fluid filament, the level lowers as the sides are approached by a quantity sometimes equal, sometimes unequal. The velocity also of the different parts of the current is large, and their respective elevation is considerable.

According to Dubuat, the principle which is exemplified in such a form of fluid surface, is as follows:—If, from any cause whatsoever, a fluid column comprised in an indefinite fluid, or contained between solid walls, should move with a given velocity, the pressure which it exercises laterally by its own motion against the surrounding fluid, or against the solid wall, will diminish that pressure which is due to the velocity with which it moves. (Principes, § 453.) Consequently, the molecules of the fluid threads, as also the surrounding ones, moving more quickly than those near the borders, exercise against them a feebler pressure, and a very large number or a higher column of them would be required to make an equilibrium. On this principle, however, philosophers are not generally agreed.

358. Various methods have been devised by different experimenters for measuring the velocity of running water, either at or under the surface. The velocity of the surface may be simply and directly found by means of a float, made of any substance the specific gravity of which is the same or a little heavier than that of water. No part of the body must, if accuracy is required, be exposed to the atmosphere. If the current flows regularly for a considerable distance, the float will be carried uniformly with the stream; and if we count the number of seconds elapsing between the float passing the extreme points of the interval, we know at once the rate of velocity per second, from the expression for uniform motion.

359. If to this float, which is supposed to be under the surface, we attach by a silken string a body specifically lighter than water, as a ball of pith of elder, part of the surface of which will be exposed to the atmosphere, and place both bodies in the middle of a current, it will be found that the pith-of-elder ball will be a little behind the one under the surface; which shows that the threads of fluid that flow fastest are not on the surface, but at a small distance, or rather immediately under it. This velocity, however, may be called that of the surface.

360. Bossut's wheel (Traité d'Hydrodynamique, § 665), similar to that in fig. 64, will also enable us to determine the surface velocity of running water. It is not, however, so correct an instrument as the float already described (358), for the friction of the screw, the resistance of the atmosphere, and the weight of the wheel, enter as elements of resistance. The small wheel WW should be formed of the lightest materials. It should be about 10 or 12 inches in diameter, and furnished with 14 or 16 float-boards. This wheel moves upon a delicate screw aB, passing through its axle Bb; and when impelled by the stream it will gradually approach towards D, each revolution of the wheel corresponding with a thread of the screw. The number of revolutions performed in a given time are determined upon the scale mm, by means of the index Oh fixed at O, and of the scale, when the shoulder b of the wheel is screwed close to a. The parts of a revolution are indicated by the bent index mm, pointing to the periphery of the wheel, which is divided into 100 parts. When this instrument is to be used, take it by the handles C, D, or when great accuracy is required, make the handles, regarded now as pivots, rest in grooves properly supported; and screw the shoulder b of the wheel close to a, so that the indices may both point to 0, the commencement of the scales. Then, by means of a stop-watch or pendulum, find how many revolutions of the wheel are performed in a given time. Multiply the mean circumference of the wheel (or the circumference deduced from the mean radius, which is equal to the distance of the centre of impulsion or impression from the axis BB), by the number of revolutions, and the product will be the number of feet through which the water moves in the given time. (Ferguson's Lectures.) Dubuat used a fir wheel of this kind very successfully. (See his Hydraul., § 441.)

361. Another instrument by which we can determine the surface velocity of water is the pendulum, which consists of an ivory or hollow metal ball fixed to a thread, the other end of the thread being tied to a graduated quadrant. The quadrant is placed above the surface, but the ball is under the water; the force of the current drags the ball with it, and the string receives a certain inclination, the fixity of the machine itself balancing the force of the current. The surface velocity, as deduced by this instrument, is equal to the product of the square root of the tangent of inclination, and a constant, or $v = n \sqrt{\tan i}$, where $v$ is the velocity required, $n$ the constant, and $i$ the inclination of the thread to the vertical. Venturi mentions this and other pendulums. Zendrini showed with this pendulum that the velocities in different parts of the section of the River Po were as the square roots of the heights, when the velocities are not very great. This has been confirmed by many succeeding experimenters.

362. To measure the velocity of running water below the surface, the tube of Pitot is the simplest instrument. Pitot's tube is one of glass, bent nearly at right angles, and expanding in a funnel-shape at the lower end, which is to be plunged into the water to the required depth. If the water be in a state of rest when the instrument is plunged into the water, the orifice being right against the current, then the level within and without the tube will be the same; but if there be a current in the water, a column of fluid will ascend the vertical part of the tube, and be a counterpoise to the force of the stream, which impels the column upward. It is natural to suppose that if the depth of the layer the velocity of which is required, be h the fluid column, the velocity will be expressed by $v = \sqrt{2gh}$. But this is not the case, for the pressure exercised on a body plunged in water is dependent on its form, and further, the pressures of the divers... Experiments on the Motion of Fluids.

The whole apparatus, by means of the ring N, slides down a long pole to the required depth, and is there fixed by the screw S; the pole D being made fast in the river. The long rod D being then in the river, and the mill fixed at the proper part of its length, and the beam in the direction of the stream, the water will in the course of a few seconds produce a uniform motion of the fins, and no sooner has this taken place than the rod E is drawn up, which necessarily lifts the toothed wheels, and places one of them in connection with the endless screw of the beam, i.e., with the fins. The apparatus is now allowed to be at work for a certain definite time, as one second, when the rod E is then let down, and the toothed wheels necessarily fall, and are caught by the projecting pieces A, A, which prevent them from turning after the communication has been stopped. The instrument is then drawn up, and the positions noted that the points A, A hold on the wheels, which will denote the number of teeth that the wheel to the right has turned round during the time of the experiment; and this number will indicate the turns that the wheel has made; for, as the beam turns round once, so does the screw, and one turn of the screw is equivalent to one tooth of the wheel passed over.

If we neglect the small amount of friction in this apparatus, the velocity of the stream is proportional to the velocity of the fins, and this last to the number n of turns which the wheel makes in a unit of time; or to N, the revolutions it makes in T seconds of time,

\[ v = \alpha \cdot n = \frac{a}{T} \]

where \( a \) is a constant coefficient for the same mill, and to be determined by experiment. We can determine \( a \), by counting the number of turns which the wheel makes in a given time when it is placed in a current of a known or uniform velocity, or by moving the instrument with a given velocity through a mass of still water; then, on dividing the number of turns by the time, we obtain the number of turns in a unit of time, and this being the divisor of the known or uniform velocity, will give \( a \). Since \( a \) is thus known, and is a constant quantity for the same mill, it will suffice to count the number of turns the mill makes to divide it by the time taken in revolving, and let this quotient be multiplied by \( a \), the product of which will give the velocity required.

The rapidity with which the mill turns under the action of a current, the velocity of which is 1 metre per second, depends on the dimensions and disposition of the fin. The palettes are generally thin, square copper plates, 0.025 met. in the side, and their radii 0.05 met. in length; the angle which their plane makes with the axis M is 45°.

365. These stream-measurers show us the curious fact, that the velocities vary at different points in the same transverse section; that the velocity diminishes, as we have already said (330), on approaching the bed, and its sides. The law of this diminution must be stated. The doctrine, however, was only taken up by philosophers after the year 1730; for, previous to that time, the common belief was that advanced by Guglielmini (in his work entitled La Misura dell' Acque Correnti, published about 1695), who supposed that the velocity at any depth of a river was as the square root of that depth. He regarded every point in a fluid mass as tending to move with the same velocity as it would do on issuing from an orifice; and, therefore, he inferred that the layer which has the greatest velocity must be that at the bed or bottom, and the layer of least velocity that at the surface, besides a continual acceleration from the river as it moves. But Pitot, in 1730, from a series of experiments on the River Seine, which he had made with his tube (362), showed that the velocities followed a law the very opposite of that advanced by Guglielmini; that they diminished from the surface on nearing the bed. Experiments on the Motion of Fluids.

Brünings, in 1789 and 1790, Ximenes, and other hydraulicians, undertook series of experiments to determine the law of diminution of the velocity. Brünings' were confined to branches of the Rhine traversing Holland; at each station he took, by the Tachometer which he had constructed for this purpose, the velocity at the depth of every foot. From these results—which may be seen in the following table—

| Names of Rivers | Depth | Velocity of Surface | Mean | Ratio of mean to surface velocity | |-----------------|-------|---------------------|------|----------------------------------| | Waal | Met. | 1:57 | 0:670| 0:827 | | Waal | Met. | 1:57 | 0:708| 0:664 | | Lower Rhine | Met. | 1:88 | 0:874| 0:779 | | Lower Rhine | Met. | 2:01 | 1:001| 0:926 | | Higher Rhine | Met. | 2:51 | 1:067| 0:938 | | Isel | Met. | 2:82 | 1:293| 1:216 | | Isel | Met. | 2:82 | 1:299| 1:243 | | Lower Rhine | Met. | 2:82 | 1:307| 1:259 | | Waal | Met. | 3:45 | 1:025| 0:938 | | Lower Rhine | Met. | 3:45 | 1:379| 1:330 | | Higher Rhine | Met. | 3:76 | 1:307| 1:220 | | Lower Rhine | Met. | 3:76 | 1:397| 1:286 | | Lower Rhine | Met. | 3:76 | 1:416| 1:361 | | Lower Rhine | Met. | 3:76 | 1:433| 1:369 | | Lower Rhine | Met. | 4:08 | 1:484| 1:341 | | Waal | Met. | 4:30 | 1:184| 1:068 | | Waal | Met. | 4:30 | 1:226| 1:131 | | Higher Rhine | Met. | 4:32 | 1:457| 1:332 |

Woltmann inferred that on reckoning from the surface the velocities diminished as the ordinates of a reversed parabola. Funk, considering the same table, supposed that the velocities followed the logarithmic curve, that is to say, that as the depths increase in arithmetical progression, the velocities diminish in geometrical progression. But M. Rancourt, from a series of experiments on the Neva, at St Petersburg, has represented the velocities on the same vertical by the ordinates of an ellipse, the smaller axis being vertical, the lower extremity being under the bottom, and the upper a little under the surface. M. Defontaine, however, from experiments on the Rhine with Woltmann's mill, gives the law of the diminution of velocities in the following terms—In general, and in proportion to the depth under the surface, in a river, the velocity of the water gradually diminishes, at first insensibly, then more and more perceptibly, and increases very rapidly on nearing the bottom, where the velocity is almost always more than half the velocity at the surface. The curve may be easily represented by laying off on a vertical, lengths proportional to the depths, and horizontal lines proportional to the corresponding velocities of the table. The curve approaches nearly to an arc of the parabola, the ordinates of which are the velocities, diminishing by a constant quantity.

These experiments of M. Rancourt on the Neva, and those of M. Defontaine on the Rhine and its affluents, are among the most important made on rivers in modern times.

The object of M. Rancourt was to ascertain how far the law of velocities of water in pipes coincided with that in open channels, when a river, for example, was frozen, and when free from ice. During the winter, therefore, of 1824, when the Neva was frozen, a part of it was chosen, which was 900 feet wide, and 63 feet deep, with a section very regular, so that this part of the river was, as it were, an immense pipe. Several holes were now made in the ice till the water was reached, and an instrument like the compass ship's log sent down; the maximum velocity was found to be a little below the centre of each vertical, and diminished as either bank was neared. After repeated trials, the relative velocities were found to differ only by 4th from each other.

The greatest velocity was found to be a little below the centre of the deepest vertical, and = 2 ft. 7 in. per second;

Velocity near the top, = 1 ... 11 ...

Velocity near the bottom, = 1 ... 8 ...

During the summer of 1826 Rancourt commenced anew his experiments on the same river, both in calm and windy weather: when calm, the maximum velocity was the same as the surface velocity; when windy, the acceleration was greater or less. Detrem and Henry give, in the Journal des Voies de Communication for 1826, an account of the experiments which they undertook for the purpose of verifying the results of Rancourt. They found that the ratio of the mean to the superficial velocity was as 0:715 : 0:903; that the maximum velocity diminished from the upper to the lower part of the river from 1:79 to 1:015 met.; and that the volume of the Neva per second was 3284:54 cub. met. The inclination also was found to be 1 in 37078.

367. The object of M. Defontaine's experiments was the execution of certain works which were intended to restrain and regulate the course of that part of the Rhine adjoining the French territory. That part of the Rhine to which Defontaine mainly directed his attention, is comprised between Basle and Neubourg. The river here is very irregular in its motion, owing to the islands and sand-banks; its inclination also is very variable. The greatest slope is near Basle, at low water, owing to the rocks; during times of floods, the inclination decreases by four-fifths its value.

In the mean state of the Upper Rhine the inclination is 1 in 1037; while at the frontier below the confluence of the Lauter, after a course of 224:160 met. along the French shore, it is 1 in 2534, or nearly a third part of the former. The total fall being taken at 143:935 met., the mean inclination is 1 in 1545, which is nearly that of the river at Brisack and Sponeck. Besides the valuable information respecting this river which is to be found in Defontaine's work, Traeux du Rhin, we have the velocities of the river at different parts of its course.

The following table shows that the decrease of velocities is irregular, and that they do not follow the law of the square roots of the inclinations—

| Names of Places | Velocities per second | |-----------------|-----------------------| | Basle | Low Water: 1:65 | | | Mean Water: 2:25 | | | High Water: 4:16 | | Huningen | Low Water: 1:70 | | | Mean Water: 2:75 | | | High Water: 4:16 | | In the angle of Krembs | Low Water: 1:88 | | | Mean Water: 2:62 | | | High Water: 4:16 | | Schalampe | Low Water: 2:67 | | | Mean Water: 2:79 | | | High Water: 4:16 | | Opposite Vieux Brisack | Low Water: 1:81 | | | Mean Water: 2:15 | | | High Water: 3:60 | | Sponeck | Low Water: 1:92 | | | Mean Water: 2:87 | | | High Water: 4:16 | | At Artohheim | Low Water: 1:97 | | | Mean Water: 2:51 | | | High Water: 4:16 | | At Guerzheim | Low Water: 2:19 | | | Mean Water: 2:93 | | | High Water: 4:16 | | At the Bridge of Reichenau | Low Water: 1:99 | | | Mean Water: 2:73 | | | High Water: 4:16 | | At Offendorf | Low Water: 1:40 | | | Mean Water: 2:25 | | | High Water: 4:16 | | At Drusenheim | Low Water: 1:49 | | | Mean Water: 1:97 | | | High Water: 4:16 | | At Belheim | Low Water: 1:24 | | | Mean Water: 1:73 | | | High Water: 4:16 | | Limit of Bavarian frontier | Low Water: 0:97 | | | Mean Water: 1:56 | | | High Water: 4:16 | | At Mannheim | Low Water: 0:70 | | | Mean Water: 1:20 | | | High Water: 2:30 |

368. M. Defontaine made also a series of experiments with Woltmann's mill on the velocities of the Rhine in different sections; in the determination of which an extent of 60 met. was traversed, and a boat was used, so suspended that its specific gravity did not exceed that of the water. The results are seen in the following table:

---

1 Annales des Ponts et Chaussees, tom. iv., p. i., 1832. 2 ib., tom. vi., 1833. Defontaine deduces from these results,—1st, That the greatest velocity is at the surface; 2d, That the velocity, diminishing at first insensibly, decreases rapidly towards the bottom, in a ratio dependent on the nature of the bed; 3d, That supposing two right lines to pass through the extremities of four ordinates, determined by experiment, and conveniently chosen in the curve, which should pass through all the points obtained, the ordinates of these right lines, corresponding to the velocities observed in the other points, will differ little in the numerical expression of these velocities; 4th, That the point of intersection of two right lines, which each partial surface of partial motion circumscribes, has for its ordinate a numerical value, which differs very little from the mean velocity expressed by the quotient of the surface of motions divided by the depth of water; 5th, That the mean velocities resulting from the preceding observations are greater than the mean velocities adduced from the velocity of the surface by means of the formula adopted for gauging streams: 6th, That the position of the ordinates, which expresses the mean velocity of each surface of partial motion, is nearer the bottom than the surface, or ¼ds of the depth, reckoning from the surface, and half the depth when the bottom is very regular (Remnie's Reports).

369. It appears from Brünning's table (365) that there is a gradual though small diminution (about ¼th) between the surface and mean velocity; the relation ranges between 0·98 and 0·96. Ximenes found this relation from his experiments on the Arno to be 0·92; and Defontaine, from his experiments on the Rhine, about 0·87. For great rivers, however, the relation is a little above 0·90. In all the preceding tables, which contain the mean and surface or maximum velocity, the former will be found to be nearly equal to ¼ths of the latter; and conversely, by adding ¼th or so of the mean to itself we find the surface velocity.

370. In gauging a river, a station is selected, and soundings of the vertical transverse section at that point are taken. The transverse area is divided into partial areas, each of which is calculated. Then by means of a boat and Woltmann's mill, the mean velocity at each of the places where soundings are taken is deduced from five, six, or seven direct experiments. The mean of these mean velocities gives the mean velocity of the river, which, multiplied by the sum of the partial areas, or the whole area of the trapezium, will give the waste of the river.

In order to find an approximate gauge a station may be selected on the river, the bed of which, for some hundred yards, is very regular. From a few soundings the area of a transverse section is determined. By means of a float, the velocity of the fluid thread in the length taken is found, and consequently of that in the measured section. M. Prony's formula (331) furnishes us with the mean velocity of the section, the product of which, with the area, will be the discharge required.

371. The following table illustrates pretty correctly, by its velocities and inclinations, the nature of the rivers of Great Britain, which frequently in their course embrace nearly all the eight descriptions:

| Characteristic of Rivers | Velocity in feet per minute | Inclination in inches per mile | |--------------------------|-----------------------------|------------------------------| | 1. Channels where the resistance from the bed and other obstacles equal the current acquired from the declivity, so that the waters would stagnate were it not for the impulse of back water | 50 to 120 | 2-00 to 5-28 | | 2. Rivers in low flat countries, full of turnings and windings, and of a very slow current, subject to frequent and lasting inundations | 60 | 12-18 | | 3. Artificial canals in Holland, Dutch and Austrian Netherlands | 30 to 40 | 2-00 to 9-05 | | 4. Rivers in most countries that are a mean between flat and hilly, which have good currents, but are subject to overflow; also the upper parts of rivers in flat countries | 90 | 15-84 | | 5. Rivers in hilly countries, with a strong current, and seldom subject to inundations; also all rivers near their sources have this declivity and velocity, and often much more | 130 | 19-8 | | 6. Rivers in mountainous countries, having a rapid current, and straight course, and very rarely overflowing | 180 | 24-37 | | 7. Rivers in their descent from among mountains down into the plains below, where they run torrent-wise | 300 | 31-68 | | 8. Absolute torrents among mountains | 480 | 37-27 |

SECTION X.—ON THE INFLUENCE OF HEAT ON THE MOTION OF FLUIDS.

372. In all the experiments related in this chapter, and in those of the Chevalier Dubuat, which are given in the article Water-works, the temperature of the water employed has never been taken into consideration. That the Experiments on the Motion of different Fluids, at different degrees of Temperature, in Tubes of Glass.

| Name of the fluid | Diameter and length of the pipe | Head of water above the top of the tube | Height of the water in a minute in inches | Velocity in seconds in inches | Degrees of heat above the freezing point | |-------------------|---------------------------------|----------------------------------------|------------------------------------------|-------------------------------|----------------------------------------| | Rain water | Horizontal | 5'2777 | 13'057 | 3 | | | Salt water | Horizontal | 5'1686 | 12'7823 | 3 | | | Salt water | Horizontal | 5'2222 | 12'947 | 11 | | | Salt water | Horizontal | 9'25 | 25'8845 | 10 to 11 | | | Alcohol | Horizontal | 5'0000 | 7'5833 | 18'7511 | 19 | | Mercury | Horizontal | 9'124 | 9'2775 | 10 to 12 | | | Mercury | Horizontal | 9'0106 | 4'0833 | 10'1021 | 10 to 12 | | Mercury | Horizontal | 2'944 | 6'1111 | 16'3558 | 10 to 12 | | Rain water | Horizontal | 8'875 | 5'2777 | 27'455 | 55 | | Rain water | Horizontal | 15'2016 | 6'1166 | 35'290 | 30 | | Rain water | Horizontal | 15'2016 | 7'0833 | 36'847 | 36 | | Alcohol | Horizontal | 5'290 | 7'2013 | 37'461 | 56 | | Alcohol | Horizontal | 5'290 | 2'50 | 13'603 | 12 | | Mercury | Horizontal | 1'225 | 1'75 | 9'103 | 10 to 12 | | Mercury | Horizontal | 7'0822 | 3'00 | 15'606 | 10 to 12 | | Mercury | Horizontal | 5'1666 | 4'25 | 22'108 | 10 to 12 | | Alcohol | Horizontal | 9'292 | 1'125 | 10'402 | 12 |

Hence our author concludes that the velocity of water diminishes as its temperature approaches to that of the freezing point, and vice versa; that salt water has a less velocity than rain water, that alcohol runs slower than water, and mercury more rapidly.

374. The general result of M. Girard's experiments has been already given in the History of Hydrodynamics. His experiments were made with copper tubes of exactly the same internal diameter, and drawn upon steel mandrils; and he employed two sets of these tubes of different diameters. The first set consisted of tubes whose length was two decimetres, and diameter 2'96 millimetres, and they screwed into each other so as to form tubes of various lengths, from 20 to 222 centimetres. The second set consisted of smaller tubes, whose diameter was 1'83 millimetres. These tubes were then fixed horizontally in the sides of a reservoir, which was a cylinder of white iron 25 centimetres in diameter, and 5 decimetres high. The reservoir was kept full by the usual contrivances; and the water discharged by the tube subjected to trial was received into a copper vessel horizontally, whose capacity had been accurately ascertained. The filling of the vessel was indicated by the instant when the water which it contained had wetted equally a plate of glass which covered almost the whole of its surface, and the time employed to fill this vessel was measured with great accuracy. The temperature of the water was also carefully noted. The results thus obtained amounted to 1200, and were arranged by M. Girard into thirty-four tables, according to the different circumstances of the experiment. When the capillary tube has such a length, that the term proportional to the square of the velocity disappears in the general formula, the velocity with which the fluid is discharged is affected in a very singular manner by a variation of temperature. If the velocity is expressed by 10 when the temperature is 0° of the centigrade thermometer, the velocity will be so great as 42, or increased more than four times, when the temperature amounts to 85° centigrade. When the length of the capillary tube is below the above mentioned limit, a variation of temperature exercises but a slight influence upon the velocity of the issuing fluid. If the length of the adjutage, for example, is 55 millimetres, and if the velocity is represented by 10 at 5° of the centigrade thermometer, it will be represented only by 12 at a temperature of 87°. In conduit pipes of the ordinary diameter, a change of temperature produces almost no perceptible change in the velocity of efflux. M. Girard also found, that the quantity of water discharged by capillary tubes, varied not only with the fluids which were used, but with the nature of the solid substance of which the tubes were composed.

SECTION XL.—INDIVIDUAL EXPERIMENTS.

1. Experiments of M. Hachette.

375. The experiments of M. Hachette were communicated to the National Institute in 1816. The first thing that engaged his attention was to determine what effect the figure of the orifice in a wall or plate had upon the waste in a given time. Making trial of circular, triangular, and elliptic apertures, and one formed of a circular arc and two straight lines, he found that when the areas of the opening and pressure are constant, the volume discharged is the same; but when he made use of an aperture with re-entrant angles, the waste was either in excess or defect. He found, also, that when the plane in which the aperture is pierced is not horizontal, the issuing fluid vein describes a curve, which is a parabola corresponding to a certain initial velocity.

376. The ratio of the contracted area to that of the orifice was particularly attended to. It appears that the smallest contraction observed, and which answered to the smallest aperture, was 0'78 of the orifice. The diameter of the orifice was 0'039 of an inch; but for diameters in excess of 0'39 of an inch, the contracted section was nearly constant, ranging between 0'60 and 0'63. The contraction varied a little with the altitude of the fluid, but was independent of the direction of the jet.

Another result reached by M. Hachette was that, all things remaining the same, the waste is a minimum when the surface in contact with the fluid is convex; when the surface is plane it increases, and is a maximum when the surface is concave. The waste was found to vary about 1/3rd, according as one or other side of the plano-concave copper disc containing the orifice was turned towards the fluid; sometimes it was greater.

377. With respect to the form of the fluid vein when the aperture was a regular polygon, each side of the polygon became a base, not of a plane but of a surface, convex externally from the orifice to the contracted section. The concavity of the surface, after having reached a maximum, diminishes near the contracted section; beyond which it changes, in consequence of the velocity acquired, into a sensible convexity, so as to show a salient edge where before there was a hollow. The hollow and the salient edge succeeding it are produced on the middle of the side, and are situated in a perpendicular plane. If the orifice present in its outline a re-entrant angle, an edge, hollow at first and then convex, passes by the summit of this angle.

378. M. Hachette next made experiments on additional tubes, that is, on tubes fitted into the orifice. These were to explain the fact of cylindrical tubes in orifices giving a greater discharge than through a simple orifice. His object was to show that the attraction of the sides of the pipe was the principal if not the main element in producing this. The fluid employed was mercury, and the pipe of iron. When the mercury was pure, the flow took place as through a simple orifice, for the iron had no affinity for the fluid; but when impure, the discharge increased, for the walls were wetted. Again, when he greased or waxed the interior of the pipe, the flow was as through a simple orifice. When the discharge, however, was continued for some time, the wax got wet, the water filled the tube, and the discharge increased.

379. The phenomena respecting the discharge of a fluid into a receiver was observed by this experimenter. From his experiments on this subject under the exhausted receiver, it appears that no increase in the discharge by pipes takes place. In order to show this, a full stream flowed through a tube under the receiver of an air-pump; when the pressure of the remaining air was reduced to 23 centimetres, the external pressure being 76 centimetres, the fluid vein was seen to detach itself from the sides of the tube. As the air within was more and more rarified, the effect of the external pressure increased, and, at the same time, was transmitted to the pipe by means of the fluid contained in the vessels, add to which also the pressure of the fluid itself. But on repeating the experiment in air, he required over the orifice a column of water equal to 22°8 met. high, to detach the vein from the sides of the tube; the external pressure being still 76 centimetres. Hence, the difference, as measured by water, between the superior and inferior pressure was 22°8 - 10°33 = 12°47 met., or equivalent to 91 centimetres of mercury.

If in this rarified state of the receiver, the external air be re-admitted within it, the water does not begin to flow again in a full stream as it did at first; the detachment of the vein from the sides of the pipe continues, as it did under the exhausted receiver, although the full pressure of the atmosphere is now acting on the water.

From experiments of this nature Hachette concluded that adhesion of the fluid to the sides of the tube takes place only at the commencement of the motion, before the fluid has acquired a sensible velocity in a direction which separates it from the side. He verified this in the following manner:—He allowed water to flow in a full stream by a pipe outside the receiver of an air-pump, when on making a small hole in the pipe near the orifice, the external air entered the pipe, interposed itself between the fluid and the sides of the pipe, the contraction took place inside the tube, and the water ceased to flow in a full stream. The little hole was now shut, but the adhesion of the water and the pipe was not again produced, the flow taking place as if the pipe had been removed.

380. Besides making experiments on long and short tubes emanating from the side of a reservoir, he took the case of a pipe penetrating the orifice inside the vessel containing fluid; and found that when the inserted end of the pipe was very thin, the effect was the same as when the orifice was made in a plate convex towards the inside of the vessel, the waste by such a pipe necessarily diminishing.

381. In order to show that the contraction of the vein diminished with the height of the fluid, he made an orifice 27 millimetres in diameter, and, under a pressure of 15 centimetres, obtained a contraction of about 0°40, or a contracted section of 0°60; while the same orifice, under a pressure of 16 millimetres, gave only a contraction of 0°31. Hence, since the vein contracts as the pressure increases, the vein in passing through a pipe may be made to detach itself altogether from the sides of the pipe, as has been already stated.

382. When the pressure or head of water over the orifice is very small, the fluid vein assumes a particular form, very different from what it had before, and Hachette came to the conclusion that it was independent of the outline of the orifice. He observed these particular forms with orifices and pipes of all figures and sizes; they go by the name of secondary veins.

383. Besides experimenting on water, Hachette employed other fluids passing out of orifices and pipes. The phenomena of mercury he found to be the same with those of water. Alcohol, a more volatile fluid than water, flows out more readily; and, consequently, a smaller pressure is required to detach its particles from the walls of the pipe than for those of water. When oil was used, its viscosity increases very considerably the duration of flow through small orifices. In the case of an orifice 1 millimetre broad, the time of flow was three times greater than for water. The discharge, indeed, by a fine tube a millimetre broad, and 5 centimetres long, was, with oil, drop by drop, but when water passed through the same pipe there was a fine stream. The head of oil was 1 met. high.

384. Hachette also made experiments to show the effect of the medium in which the experiments were made; how it modified the pressures on the orifice when a flow took place; and how it opposed a certain resistance to the discharge or motion of the fluid. He also experimented on the impediment to the motion of water arising from the fluid vein impinging on fixed planes, placed at different distances from the orifice. A vein issued out of a large vessel by a circular horizontal opening 20 millimetres in diameter, and was received in a vessel at some distance from the orifice. The level of the fluid in the vessel was depressed about 6 decimetres in 10° 20'. The plane face of an obstacle at different distances from the orifice was then presented to the jet impinging perpendicularly.

The distances in millim. were 128, 80, 50, 24, 4; the times in sinking six decim. were respectively 10° 21', 10° 25', 10° 26', 11° 13', 15° 24'; which shows that, at the distance of 128 millimetres, or 5 inches, the obstacle presents no sensible effect; but at 4 millimetres, or 0°157 of an inch, the time is increased rather more than one-half.

2. Experiments of Dr Matthew Young relative to the discharge of fluid through additional tubes.

385. Dr Young's paper containing an account of these experiments will be found in the 7th volume of the Transactions of the Royal Irish Academy. The most interesting part of the memoir is that relating to the increased discharge through additional tubes. They were performed about the year 1798.

When a tube, mmrs (fig. 66), is inserted into the vessel ABCD, it is found that the velocity is increased nearly in the subduplicate ratio of the length of the pipe when the tubes are short; and that it approaches nearer to that subduplicate ratio, according as the length of the pipe is increased. The explanation of this increase of velocity is somewhat difficult, since the water cannot issue at rs with a greater velocity than it enters the tube at mm; and it does not appear how the velocity of the fluid can be increased by the insertion of a tube in its bottom. In order to explain this, we must consider that the weight pressing down the plate... mn is the weight of a fluid column enm, together with the atmospheric column of the same sectional area, while the upward pressure on the plate mn is a corresponding column of air, less the weight of a fluid column mnr; hence the plate mn is pressed down by the weight of a fluid column efr. The velocity, then, with which the fluid lamina mn will issue through the orifice mn will be the same as through the orifice rx in the vessel ABD; or that which a heavy body would acquire in falling down through the vertical height er; and the same will be the case for all the laminae between mn and rx; for they cannot descend faster, since otherwise a vacuum would be left in the tube, which cannot take place owing to the upward pressure of the atmosphere. The velocity of escape will be the same, whatever be the pressure of the atmosphere, provided only that the weight of a column of air, of a base rx, and a height equal to that of the atmosphere, be either greater than or equal to the weight of the fluid cylinder mnr. This might be proved experimentally by inserting a pipe in the bottom of a vessel of water or mercury, and the whole placed under the exhausted receiver. Let x be the defect of the measure from the standard altitude, it will determine the pressure of the air on the mercurial surface; let also y be the height of the mercury in the vessel above the upper orifice of the pipe, and l the length of the pipe, then the whole downward pressure on the plate of mercury which is immediately in the upper orifice of the pipe, is \(x + y\); and the upward force pressing the same plate or layer of mercury is \(x - l\): the difference between these two quantities or forces equals the absolute force pressing that plate downwards, and will be expressed by \(y + l\) so long as \(x\) is greater than \(l\); if \(x = l\), then \(x - l = 0\), and the force pressing the plate downwards is \(x + y = l + y\); hence, therefore, no variation in the time of efflux will be perceived while the height of the mercury in the gauge or measure is equal to or less than the difference between the length of the pipe and the standard altitude. Let \(x\) be less than \(l\), then there is no upward force; and as before the force pressing the plate downward is \(x + y\), which varies as \(x\) when \(y\) is constant, and which will therefore decrease as \(x\) decreases; if \(x\) vanish or is equal to nothing, or when the receiver is absolutely exhausted, the force is \(y\), and the time of efflux will be the same as if the pipe had not been inserted in the bottom of the vessel.

386. In order to test his theory, Dr Young inserted a tube 7½ inches long, in a cylindrical vessel, closed the orifice of the pipe, and filled the vessel to a depth of 6 inches with mercury; he then placed the whole apparatus under the receiver of an air-pump, the barometer being at 30 ins., and the gauge at 28½; the time of efflux was found to be 26 seconds; when the experiment was repeated under the same circumstances, but in the open air, the time of efflux was found to be 19 seconds. Now, since the gauge stood at 28½ inches, the defect \(x\) was \(30 - 28\frac{1}{2} = 1\frac{1}{2}\); and as the pressure on the mercurial plate was \(6 + 1\frac{1}{2} = 7\frac{1}{2}\), but in the open air was \(6 + 7\frac{1}{2} = 13\frac{1}{2}\), therefore the ratio of the velocity of efflux in both cases, which is the same as the inverse ratio of the times, was as \(\sqrt{7\frac{1}{2}} : \sqrt{13\frac{1}{2}} = 2\frac{1}{2} : 3\frac{1}{2} = 19 : 26\) nearly. The times of efflux were the same in the open air and under the exhausted receiver, except when the gauge stood higher than 22½ inches; i.e., unless the height of mercury in the gauge exceeded the difference between the length of the pipe and standard altitude. In another experiment which he made, the gauge standing at 27½, and the barometer at 29½ inches, the effect was found equal to 2, and the pressure was 8. But \(\sqrt{8} = 2\cdot828\), and \(\sqrt{13\frac{1}{2}} = 3\cdot71\), now \(2\cdot828 : 3\cdot71 = 19 : 24\) nearly; by experiment the time of efflux was found to be 23 seconds. When the efflux is made in vacuo, the pipe is not filled during the efflux, as it is while the discharge is made in the open air.

Since the water column in pipe mnr adds to the pressure which the plate mn sustains, by diminishing the upward pressure of the air through the pipe, it appears that it produces this increase of pressure in the plate mn alone, without producing any lateral pressure in the water on a level with mn; for, by inserting a pipe in the side mb or mc, the velocity of escape would not be affected, and, therefore, the plate mn, immediately over the orifice of the pipe, is the only one on the same level the tendency of which downwards is increased by the insertion of the pipe. Therefore, the fluid particles of the edge of the aperture will issue through the orifice mn perpendicularly, since they have their vertical pressure increased by the weight of a column mnr, while the lateral pressure is unchanged; the sides also of the tube mnr will prevent the converging tendency of the particles, and, therefore, on both accounts, the volume of water discharged through a pipe thus inserted, will exceed that discharged through a simple orifice in a ratio greater than the subduplicate of the height of the water. If the pipe be lengthened, the ratio of the volume actually discharged to that determined by theory, will also increase, since the ratio of the perpendicular to the horizontal pressure increases in the ratio of the sum of the depth of the vessel, and length of pipe to the depth of the vessel. Hence in experiments of this sort, results will more nearly coincide with those of theory than when made with a simple orifice, except when the tube is so long that the friction of its sides will retard the fluid; or when so short that the particles have not acquired a vertical direction. All this agrees with the experiments of Mr Vince, an account of which is given in the Phil. Trans. for 1795. Mr Vince, e.g., inserted a tube a fourth of an inch long into a cylindrical vessel twelve inches deep, and found that the velocity did not sensibly differ from that through the orifice; the cause of which was that the stream did not fill the pipe, but that the vein contracted as if it flowed through a simple orifice; when the pipe was half an inch long and inserted into a vessel of the same depth, the velocity from this pipe and from the orifice was in the ratio of 4 : 3 nearly; whereas, by theory, it is \(\sqrt{12\frac{1}{2}} : \sqrt{12} = 4\frac{1}{2} : 4\) nearly. Now, if 4\(\frac{1}{2}\) : 4 be increased in the ratio of 7 : 6, since the latter is the ratio of the diminution of velocity, and the contraction either nearly or entirely vanishes in a pipe, it will give the ratio of 3\(\frac{1}{2}\) : 3. When the pipe was one inch long, the ratio of the velocity from the pipe and from the orifice was as 4 : 3 nearly; whereas, by theory it is expressed by \(\sqrt{13} : \sqrt{12} = 26 : 25\); and if 26 : 25 be increased in the ratio of 7 : 6, we have the ratio of 3\(\frac{1}{2}\) : 3. When Mr Vince employed longer pipes, the velocity of efflux by experiment approached nearer to that which ought to have been discharged by theory; hence, in long pipes, he says, the difference between theory and experiment was not greater than what might have been expected from the friction of the pipes, and other retarding causes. On inserting a pipe of the same diameter with the aperture which terminated in a truncated cone fixed in the orifice, he expected from this arrangement, that the volume discharged in a given time would be less than otherwise, since the water issuing through the orifice would have room in the enlarging cone to form the vena contracta; but the result was, that the same volume was discharged as if the pipe had continued throughout of the same diameter with the orifice. The reason of this is, that the pressure of the air will not allow the truncated cone to remain partly empty, as it would if the vena contracta were formed; hence, it continues full, and, therefore, the water will pass through it in the same manner as if the water in the cone surrounding the pipe were congealed.

387. Mr Vince next inserted into the bottom of the vessel a perpendicular pipe in the form of a truncated cone, the narrow end being in the orifice, and found that the efflux increased more than if he had inserted a cylindrical pipe of the same length, the diameter of which was equal to the narrowest section of the conical tube. Mr Vince explains this on the same principle as that by which we account for the swelling of the diameter of a vertical fluid vein, flowing through a simple orifice with a considerable velocity. For when a vertical pipe is inserted, the velocity of discharge being increased, the resistance from the air will be so also, and thus the diameter of the vein tends to enlarge itself.

Now, the truncated cone pipe, the wide end outwards, is just a pipe to admit of this thickening or swelling of the vein; and the water in escaping discharges a larger volume, and with an increased velocity than when through a straight pipe; whereas, in the cylindrical pipe, the fluid cannot enlarge itself, and, thus confined, the efflux must be retarded, and the volume discharged in a given time diminished. Accordingly, under the receiver of the air-pump, with even a moderate degree of exhaustion, there is no difference observed between the velocities with which a fluid is discharged through a conical or cylindrical pipe.

Dr Young made also several experiments on the contraction of veins, for an account of which see vol. ii. of the Irish Academy.

3. Experiments of Poncelet and Lesbros.

388. The experiments of Poncelet and Lesbros were made at Metz, in the years 1827 and 1828. The series was a very extensive one on the expenditure of water through rectangular orifices of considerable dimensions; and as they were undertaken by order of the French government, no expense was spared in making them as complete as possible. The objects which the experimenters had principally in view, were to ascertain the exact measure of the coefficient of contraction, and to determine the forms of the fluid veins under different charges and areas.

389. In order to get a correct form of the vein, several veins were projected on the plane of the orifice. These projections were taken by means of an iron rod, having over nearly its whole surface, threads of a screw, and ending at its lower extremity in a delicate point, which was put successively in contact with every point of the exterior surface of the jet. The screw worked perpendicularly into a wooden nut, and which, when the screw was acting, glided along one side of a rectangular or octagonal frame of wood, placed between the orifice and the rena contracta.

The orifice that especially claimed their attention while taking these projections, was a square of 0'2 met. in the side, and the mean charge over the centre was 1'68 met. This orifice was selected, because the jet spouted with much stability, its dimensions were such as to admit of manipulating very accurately, and because of its distinctness. Projections, then, of the jet from this orifice were taken first of all at a distance of 0'05 met. from the plane of the orifice; but they confined themselves to the space between 0'064 met. and 0'50 met., since the former was too near, and the latter was just at the limit where the chances of error begin to multiply. At the distance of 0'05 met., however, the jet was perfectly stable. The error in taking the projections did not exceed a millimetre. Two of those veins projected are represented in fig. 67. ACEG is the square orifice of 0'20 met. in the side, and under the charge 1'68 met.; at the distance of 0'15 met. from the orifice, the section of the vein was abedegfh; but at the distance of 0'30 met. the jet had the form represented by abedegfh'. This last, of all the nine projections taken, had the least section; its area was that of the orifice as 0'562 : 1; while the ratio of the real to the theoretic waste was 0'605 : 1; they would have been equal, if the velocity at this the least section had been that due to the height of the reservoir. The particles b', d', f', h', in the plane of the section of the vein, are those which issue from the middle points, B, D, F, H, of the side of the square. The section, also, of the vein is apparently a kind of square, the angles of which correspond to the middle points of the sides of the orifice, or that it has turned round part of a revolution. The vein abedegfh made an eighth of a revolution. A phenomenon of this kind taking place in a vein issuing from an orifice not circular, is known by the renversement, the turning or inversion, of the fluid vein.

390. In the course of experiment it was observed, that as the height of the fluid in the reservoir diminished, the phenomenon of the vein turning also diminished; that is to say, that the mass of particles all round the central axis gradually approached that axis, more so on the upper part of the vein, for gravity acting on all the fluid particles of the vein bends the whole mass, especially the horizontal sheets. Poncelet and Lesbros could not observe, as Hauchette and other experimenters had done, spiral fluid threads crossing each other at certain angles on the surface of the vein, nor the successive swellings and shrinkings of Savart and Bidane. The former they attributed to very fine ridges on the lower edge of the orifice; and the reason why the latter were not seen was owing to the small extent of vein observed.

391. With respect to rectangular orifices, where the breadth exceeds the height, the turning of the vein was less distinct than for the square orifice, and still less when the height of the reservoir decreased. In the case of an orifice of a centimetre of an opening, the angles and jutting parts were scarcely perceptible, which shows that the greatest contraction tends to take place at a great distance from the orifice. This vein presented the appearance of a very thin liquid plate, which continually shrunk in its horizontal, but thickened in its vertical direction, indicating that the liquid masses belonging to the angles bordering the outline of the orifice, tend here naturally to reunite into one and the same mass.

In order to explain this turning of the vein, they suppose that those particles which in the plane of the orifice form the centre of the vein, come principally from parts of the reservoir bordering the prolongation of the horizontal axis of this orifice, and describe paths sensibly parallel to this axis; while those particles belonging to the other parts of the orifice come from points of the reservoir as distant from this axis as they belong to points more remote from its centre; and hence the particles not only come into the plane of the orifice under inclinations greater and greater with respect to the axis of the vein, but they come in greater abundance.

392. The following table will show the contractions which a vein experienced at different distances from the plane of the orifice, the areas of the different sections being deduced by means of Simpson's Method of Quadratures. The orifice was a square of twenty centimetres, under a charge of 1'68 met. It appears from the table, that the maximum of contraction of the section takes place at nearly thirty centimetres' distance from the orifice, or nearly 1'5 times the side of the orifice; but beyond this distance the contraction continually diminishes, either because the vein dilates over its whole extent, or because its profile becomes more and more oblique with respect to the central thread. The amount of error does not exceed the 150th of the true value, and hence the maximum contraction may be taken equal to 0'437 of the area of the orifice, and the corresponding coefficient as 0'563. This coefficient is sensibly equal 393. In order to show the variation in the value of the coefficient for different charges and for different orifices, curves were traced wherein the coefficients were regarded as ordinates, and the charges as abscissae of a curve constructed for each orifice, and by its aid the ordinates were determined; that is, the coefficients intermediate to those directly determined by experiment. These curves followed, in general, a very regular motion, although distinct for each orifice; and their continuity was somewhat perfect, for they were subjected to pass through points deduced from experiment. The nature of these curves was shown by Lesbros to be one of a parabolic kind,—the real wastes, however, instead of the coefficients, being substituted for the ordinates, and the charges as the abscissa. Lesbros found also the common and equilateral hyperbola on varying the quantities representing the ordinates and abscissae. The curves relative to orifices of 20, 10, 5, and 3 centim. always presented their concavity to the axis of abscissae, and had one ordinate a maximum; whereas for those of 2 and 1 centim. of an opening, the maximum point apparently disappeared. Some of the curves presented a very remarkable appearance, indicating a peculiarity in the law of the coefficients for the orifices of 3 and 5 centim.

Other curves, again, presented points of inflection, which were not seen in those of the orifices of 10 and 20 centim., showing again that the law of coefficients was different from those of 10 and 20 centim. of an orifice. The law, then, of the coefficients is peculiar when points of inflection enter the lines, for the degree of curvature necessarily increases with the numerous and sudden alterations of curvature. The curves also indefinitely approach the axis of abscissae, which will thus be an asymptote.

394. The interpretation of the curves contains the substance of all the experiments, for they served to construct, by interpolation, a general table of the values of the coefficients, from formulae of the theoretical waste for charges over the orifices, increasing from the smallest charge of 0.0046 met. to the greatest of 3 met.

395. The following is an abstract from this general table of coefficients as drawn up by Poncelet and Lesbros. It will give one an idea of the labour and care which they bestowed upon the experiments. The coefficients were determined by the usual formula for the discharge (179), the data given being portions of the curves corresponding to the weaker and stronger charges in the reservoir; but those thus obtained are separated from the others by a transverse line. The values of the quantities D and D' are the same as in 398 following:

| Orifices shut at the upper part. | The height of the level of the water being measured | |---------------------------------|--------------------------------------------------| | Charges on the summit of orifices in metres. | From the orifice to a point where the fluid is perfectly stagnant. | Immediately above the orifice. | | Coefficients of formula D, for heights of orifice of | Coefficients of formula D', for heights of orifice of | Coefficients of formula D, for heights of orifice of | | 20 cent. | 10 cent. | 5 cent. | 2 cent. | 1 cent. | 20 cent. | 10 cent. | 5 cent. | 2 cent. | 1 cent. | | 0.0000 | ... | ... | ... | ... | ... | ... | ... | ... | ... | | 0.0046 | ... | ... | ... | ... | ... | ... | ... | ... | ... | | 0.0091 | ... | ... | ... | ... | ... | ... | ... | ... | ... | | 0.02 | ... | ... | ... | ... | ... | ... | ... | ... | ... | | 0.08 | ... | ... | ... | ... | ... | ... | ... | ... | ... | | 0.18 | ... | ... | ... | ... | ... | ... | ... | ... | ... | | 0.30 | ... | ... | ... | ... | ... | ... | ... | ... | ... | | 1.20 | ... | ... | ... | ... | ... | ... | ... | ... | ... | | 1.40 | ... | ... | ... | ... | ... | ... | ... | ... | ... | | 1.60 | ... | ... | ... | ... | ... | ... | ... | ... | ... | | 2.00 | ... | ... | ... | ... | ... | ... | ... | ... | ... | | 3.00 | ... | ... | ... | ... | ... | ... | ... | ... | ... |

396. It appears from the above table that the coefficients increase as the charges become greater, but only up to a certain point, for they then begin to diminish, although the charge be increased. In the general table it is seen that the coefficients approach an equality in each column as the charge increases.

397. Besides numerous experiments which they made on weirs, Poncelet and Lesbros endeavoured to determine the limiting case, or the point of natural transition between a closed and an open orifice; that is, between a simple orifice and a weir. In one of their experiments on weirs, where the charge over the base of the orifice was 0.217 met., the corresponding coefficient was very near that for a simple orifice. But, in order to determine it experimentally, they lowered very gently the level of the water in the reservoir, beginning at the instant when the summit of the orifice was entirely covered with water; they then opened the sluice of their canal a little more than was requisite, in order that the total quantity of water spent by this sluice and by the orifice, might be exactly equal to that which flowed into the reservoir. In this manner they easily determined the precise instant when the fluid was detached from the summit of the orifice. The operation being repeated at various times, always gave a total charge of 0.2205 met. over the base of the orifice; when the charge exceeded this, the water ceased to adhere to the upper wall in a manner sufficiently stable to obtain accurately the waste. The charge of 0.217 met. over an orifice will thus correspond to the same instant when the adherence of the water to this wall is overcome, and when the orifice tends to form itself into a weir.

398. Before closing what we have to say on the experi- ments of Poncelet and Lesbros, we append the following table, which contains several results of the experiments on an orifice 20 centim. of a base, and of various heights; we give it not only to show the care with which they conducted their labours, but also to exhibit at one view several of the tabulated results. We have selected two—the first and the last—from each of the six tables in the *Expériences Hydrauliques sur les lois de l'Écoulement de l'Eau*, &c. The headings over each of the columns refer to the following quantities:—\(l = 0.20\) met. = horizontal length common to all the orifices; \(h\) = charge of fluid on the lower fixed border; \(h'\) that on the upper variable edge; \(Q = h - h'\) = the height of the orifice; \(D\) = the theoretical waste, or volume escaped in a certain time relative to the velocity \(V\); \(D'\) = the theoretical waste on taking into account the influence of the opening; \(E\) = the actual waste as determined by observation, \(H\) = mean charge over centre of orifice \(= \frac{1}{2}(h + h')\).

| Temperature of Water | Values of Orifice 10 centimetres high, 20 centimetres wide. | |----------------------|----------------------------------------------------------| | | Values of Orifice 5 centimetres high, 20 centimetres wide. | | | Values of Orifice 3 centimetres high, 20 centimetres wide. | | | Values of Orifice 2 centimetres high, 20 centimetres wide. | | | Values of Orifice 1 centimetre high, 20 centimetres wide. |

With respect to the drops which become detached from the troubled part, their formation is not, as in descending jets, an effect of acceleration due to gravity; for the drops are detached in the same manner from a vein when it is directed upward. Savart attributed the effect to an oscillation in the fluid of the reservoir, whereby the molecules of the fluid, being pressed sometimes more or less in issuing from the orifice, moved with a velocity more or less great. These movements may be seen very nicely in experiments on the resistance that the air experiences in pipes; the air may then be seen advancing irregularly as if by undulations; the waves, on propagating themselves, accelerate and retard the velocity.

401. M. Savart, moreover, found that the waves or undulations of the atmosphere had a very singular effect on the fluid vein. When the troubled part of the vein is received on the bottom of a vessel, the sound arising from the shocks of the successive drops can easily be heard; and if this sound be in unison with a musical instrument sounding at a small distance from the vein, immediately the clear, untroubled part of the jet shortens, and sometimes disappears; the swellings of the troubled part increase, shorten, and the space separating them is greater also.

These vibrations of the atmosphere have considerable influence on a descending jet, but less effect on an ascending one; for in the latter case the transverse section increases instead of diminishes in proportion as it recedes from the contracted section, which is never so small that the effect of these vibrations may become sensible.

402. Besides these peculiarities connected with a fluid vein, Savart investigated the action of a jet against an immovable plane, and also of two directly opposing jets. When a jet is directed against a vertical plane surface, the... surface receiving the jet must be considerably larger than the magnitude of the orifice. Were it less than this, then evidently the water would turn over the edges of the plane; but that this be not the case, the area of the plane surface must be about eight times that of the orifice. It will now be seen that the liquid threads from the vein will spread out in all directions on the surface till they be parallel to it. As soon as they have attained their parallelism, the pressure of the jet against the plane is at a maximum, and can easily be measured.

If the plane be oblique upward, the water will spread itself more on the upper part than on the lower; the jet also does not press with so much intensity on the plate as in the last case.

When two equally powerful and equally broad jets directly meet each other, so that both will form a straight line, the water at the section of opposition forms a most beautiful transparent disc, which is surrounded by a circular rim, having watery spikes radiating from it in all directions.

403. (B.) Borda completely realized the case when a pipe with thin sides and entirely penetrating the wall of a reservoir, is employed for the purposes of discharge. Having taken a white-iron tube, 0'135 met. long, and 0'032 in diameter, and wholly placed within the reservoir, he produced, under a charge of 0'25 met., a fluid vein which was altogether detached from the walls of the tube, and the actual was only 0'515 of the theoretic waste. Various considerations led Borda to think that it might even be reduced to 0'50, or one-half of the theoretical discharge.

404. (C.) George Bidone of Turin, in his "Expériences sur la Forme et la Direction des Veines et Courans d'Eau lancées par diverses Ouvertures," 1829, gives us something very interesting on the phenomena of a fluid vein. Among his numerous experiments, he took a regular pentagonal orifice 0'014 met. in the side, pierced in a thin vertical copper-plate; the escape took place under a charge of 1'97 met. At a distance of 0'012 met. from the orifice, the transverse section, perpendicular to the axis of the vein, was a very regular decagon. The greatest contracted section, or first shrinking, was distant from the orifice by 0'030 met. Beyond this, however, the vein entirely changed its form; it presented, at a distance of 0'095 met. from the orifice, a system of five fluid radial blades, or spokes, symmetrically disposed around the axis of the vein; the breadth of these spokes increased up to the swelling of the vein, then it diminished, and the spokes reunited anew so as to form a second shrinking at a distance of 0'86 met. Beyond this the vein continued round but shapeless.

Besides a rectilineal pentagon, he successively substituted orifices which were regular pentagons, but in a convex and a concave disc, and also with salient and re-entrant angles like a star with five radiations, and the vein preserved always the same form, together with the same five radial blades or spokes.

When the orifice had 6 and 8 sides, the number of blades was 6 and 8; and the turning or inversion (389) of the vein was a 12th and a 16th of the circumference. When the opening was a narrow rectangle, but prolonged horizontally, at a certain distance the vein had no longer a broad vertical blade,—the turning appeared complete.

405. With respect to the phenomena of the troubled part of the vein, we see sometimes, beyond the second shrinking, the vein dilating itself anew, and dividing itself a second time into the same number of blades. These blades increase in breadth up to a second swelling, and then diminish so as to form a third shrinking, beyond which a third dilatation ensues, or a third swelling is formed, then a fourth shrinking.

Eytelwein, in his German translation of Michelotti's "Sperimenti Idraulici," gives a description and representation of the different forms of these swellings and shrinkings.

406. The main cause of the forms and turnings of veins is the oblique direction which the fluid fillets take on approaching the orifice, and this direction tends to urge them beyond the orifice. The action of these fillets on the forms of the veins is greater when the fluid issues out of a very acute angular opening; the latter opening compresses the vein more strongly than do other orifices, and hence the blades form at parts intermediate to those where they exercise their action. Lastly, the resistance of the air, and the mutual attraction of the particles, contribute to contract the blades, and to form a second shrinking.

407. Bidone, also, by numerous experiments at the Water-works of Turin, endeavoured to solve the question of augmented waste when the contracted vein was suppressed on one side. This suppression has been noticed already in 171. The following table gives the result of experiments to determine this point:

| The contraction being suppressed on one side | Portion of contraction | Coefficient | Ratio of increase | |---------------------------------------------|-----------------------|------------|------------------| | No side | 0 | 0'608 | 1'000 | | One small side | 4'620 | 1'020 | | One large side | 4'637 | 1'049 | | One large and one small side | 4'659 | 1'085 | | Two small and one large side | 4'680 | 1'119 | | Two large and one small side | 4'692 | 1'139 |

The figure of the orifice was confined to the rectangular form, the base of which was 0'054 met., and height 0'027. The plates, which were attached sometimes on one side, sometimes on another, and sometimes on two or three of the sides, were 0'067 met. long; that is, they advanced thus much into the reservoir. The flow took place under charges of from 2 to 6'88 met. The coefficients are noted in the table, wherein the last column indicates that unity has been taken when the orifice is entirely free; as the column of coefficients increase, so also it appears that a corresponding increase takes place in the discharge column. M. Bidone has deduced from his results the general expression

\[ Q = \frac{mS\sqrt{2gH}}{p} \left( 1 + 0'152 \times \frac{n}{p} \right) \]

in which \( n \) is the length of the part of the perimeter where the contraction is suppressed, and \( p \) the whole perimeter of the orifice. Since this expression gives as its maximum error a 39th part only, the formula representing the waste by a rectangular orifice, when one part of the contraction is suppressed, will be represented by

\[ Q = \frac{mS\sqrt{2gH}}{p} \left( 1 + 0'152 \times \frac{n}{p} \right) \]

408. Bidone made also experiments on circular orifices, one of which was 0'04 met. in diameter; and, by the aid of plates curved cylindrically, first 1-eighth of the contracted perimeter was destroyed, then, successively, 2, 3, 4, 5, 6, and 7 eighths. The annexed table shows the results obtained. By it we see that the numbers of the last column increase less rapidly than in the case of rectangular orifices, and the general expression is now only

\[ 1 + 0'128 \times \frac{n}{p} \]

409. After having destroyed 7-eighths of the contracted section, Bidone, to destroy it entirely, introduced within the reservoir 0.067 met. of cylindrical pipe, 0.04 met. in diameter. The coefficient which he now obtained was 0.767, and 1.285 for the ratio of increase. From the expression mentioned above, this number would be 1.128, where the augmentation is not one-half that really obtained.

He concluded, therefore, that the phenomena of interior additional tubes, in the case where the contracted section is wholly suppressed at the exterior edges of the orifice, is not of the same nature as when the contraction is only destroyed in part, however great that part may be; there is no intermediate case between them.

M. Bidone experimented also on Weirs; and the remarks which he made on this subject were among the first worthy of notice. See his Expériences sur la Dépense des Reervoirs, 1824.

5. Experiments of Mr George Rennie.

410. Mr Rennie, in the year 1830, experimented on the waste by orifices and tubes of different diameters and lengths, and at different altitudes. The area, length, and altitude were the three main elements that entered into these experiments. The apparatus consisted of a wooden cistern 2 feet square in the base, and 44 feet high. The water was maintained at a constant height by means of a regulating cock; a float with an index indicated the exact height at which the water stood in the cistern above the centre of the orifice. The orifices were in brass discs, about a 60th of an inch thick; the discs were fitted into a hole in the wall of the cistern, and closed by a valve of brass, ground truly to each of the plates. A lever opened the valve, and a chronometer noted the time. The following table contains the wastes from different-sized orifices, the vessel being kept constantly full, and at different heights:

| Height over charge | Real time in discharging one cubic foot | Theoretical time in discharging one cubic foot | Ratio of theoretical to real waste | Vena contracts | |-------------------|----------------------------------------|-----------------------------------------------|----------------------------------|---------------| | Feet | Seconds | Seconds | | | | 4 | 15 | 8-9 | 1:0-593 | Vena contracts about 1 inch beyond orifice; but the jet with angles reversed, and taking the sides of the triangle, the jet then expanded and lost its form. | | 3 | 18 | 10-3 | 1:0-572 | | | 2 | 22 | 12-7 | 1:0-577 | | | 1 | 30 | 17-9 | 1:0-598 | |

From the above table it appears that, with equal areas, the waste by different orifices, whether circular, triangular, or rectangular, is nearly the same, the increase being with rectangular orifices.

411. The next table contains the waste by cylindrical glass orifices and tubes, from 1 inch to 1 foot in length, and of different diameters, from a vessel kept constantly full, and at different heights.

It appears—1st, That the wastes in equal times from orifices and additional tubes, are as the areas of the orifices; 2d, That the wastes in equal times by the same additional tubes and orifices, under different heads, are nearly as the square roots of the corresponding heights; 3d, That the wastes in equal times by different additional tubes and ori-

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1 Recherches Expérimentales et Théoriques sur les Contractions Partielles des Veines d'Eau, etc., 1836. 412. From the preceding experiments, the mean coefficient for

Altitudes of 4 feet with circular orifices is ............... 0.621 But with altitudes of 1 foot, the coefficient is .............. 0.645 With triangular orifices at 4 feet altitude .................. 0.593 ........ 1 foot ........................................... 0.596 With rectangular orifices at 4 feet ......................... 0.593 ........ 1 foot ........................................... 0.616

413. Mr Rennie has also made experiments on the volume discharged by leaden pipes of different diameters and lengths, from a vessel kept constantly full, and at different altitudes.

The time in discharging 1 cubic foot is nearly double the time occupied by glass tubes of equal lengths and areas.

The following table shows the wastes from leaden pipes 0.5 inch of bore, the lengths varying from 1 foot to 30 feet:

| Glass tubes | Brass pipes | 1 inch long | 1 inch long | 1 inch long | |-------------|------------|-------------|-------------|-------------| | Feet | Sec. | Sec. | Sec. | Sec. | | 1 | 115 | 24.5 | 55 | 145 | | 2 | 15.0 | 28.5 | 63 | 157 | | 3 | 17.5 | 35.0 | 77 | 205 | | 4 | 25.0 | 53.0 | 110 | 297 |

The ratio of the waste by glass tubes with pipes 30 feet long is as .................................................. 1 : 4 And with brass orifices, is as .................................. 1 : 3

Hence the expenditure of water through pipes of equal diameters, but different in length, and under different altitudes, will be as follows:

The length being as 30 : 1, the wastes are as 37 : 1 ........ 8 : 1 .................................................. 26 : 1 ........ 4 : 1 .................................................. 20 : 1 ........ 2 : 1 .................................................. 14 : 1

The wastes by leaden and glass tubes are nearly the same.

The length of a pipe may be increased from 3 to 4 feet, without diminishing the discharge, as compared with the plate orifices.

414. The experiments on leaden pipes in order to show the effect of bends, are very interesting. Mr Rennie took the straight pipe, 5 inch of bore, on which the former experiments were made, and carefully bent it into one, two, and fourteen semicircular bends respectively, each bend being 7.5 inches in the semi-diameter, and two of the fourth part of a circle of 3.25 inches radius. The results are shown in the annexed table.

This table shows us that, with one semicircular and two quadrantial bends, when compared with a straight pipe of equal length and bore, the resistance is greater than that of the straight pipe by a 36th to a 70th of the latter. With fourteen semicircular and two 4th circle bends, the increased resistance varies from a 19th to a 39th of that of the straight pipe. The increased number of bends also does not increase the resistance in the ratio of the number of the bends, but only shows an increase of resistance, as compared with the four bends, of a 15th to a 35th.

415. In the case, again, where the bends are rectangular, the same dimensions of a pipe were employed, and the bends were in the form of right-angled elbows, each side being 6.75 inches long. The number of elbows varied from 1 to 24. The experiments in the following table show us that in the first three the waste diminishes as 2.5 : 1 ; and in the last, as 3 : 1 nearly. The whole number, however, indicate that with one right-angled pipe the waste differs only from that given by a pipe with twenty-four right angles, as 2 : 1; however natural it might be to infer that the diminution of waste would be greater as the number of right angles increased:

| Height over the centre of orifice | 1 right angle end of pipe | Straight pipe 15 feet long | 24 right angles | |----------------------------------|---------------------------|--------------------------|----------------| | Feet | Sec. | Sec. | Sec. | | 1 | 180 | 143 | 395 | | 2 | 214 | 164 | 465 | | 3 | 246 | 208 | 584 | | 4 | 371 | 312 | 872 |

6. Professor Magnus's Experiments.

416. In the year 1848, Professor Magnus of Berlin communicated to the Royal Academy of that city the results of his experiments on the motion of fluids. He was led to the subject by doubting the truth of an inference at which Venturi had arrived in one of his experiments relative to the lateral communication of fluids. The two main proofs in support of Venturi's opinion are those already mentioned in 262; and, as a consequence of the first experiment, he asserts that if any very mobile body—a light feather for example—be brought near a jet of water spouting in the atmosphere, the jet will impart a motion to it, and the body will be carried forward by the air. Professor Magnus applied to such a jet as delicate a test as he could conceive; he held the flame of a candle to the transparent portion of the vein, but was unable to detect the slightest motion. But on applying it to the second or third of the swellings of the vein, the flame became slightly disturbed; and at that part where the jet is broken up into detached masses it was violently agitated, and at last put out by the spray. This was the case when the jet was regular and steady; but if it oscillated under the vibrations of the air, then the flame was carried forward at an anterior part of the jet.

Now it is clear that, if the air were, in consequence of its adhering to the jet, carried along by the water, this action must be strongest where the velocity of the fluid particles is greatest, that is, near the *vena contracta*. But as this is not the case, as the flame begins to oscillate only at the time when the jet begins to be unsteady, it is very probable that if this motion in the jet were to cease, the irregularity in the flame would also vanish.

417. The other experiment of Venturi mentioned in 262, was ingeniously modified by Professor Feilitzsch (Poggen-dorf's *Annalen*, vol. lxiii., p. 216). He took a rectangular vessel EDGF, divided into two parts by a partition HI. A cylindrical pipe ABC, 2½ inches in diameter, and 8 inches long, with both ends open, and close to the bottom EF, was a means of communicating with both divisions. The supply pipe, about 6 feet long, is, by b, bent horizontally at b, and having in this extremity a jet tube ab, which spouts within ABC near BC. Suppose, then, the water to be at the same level in both divisions of DF, say to the opening K, and let the jet begin to work into the pipe ABC, then the level of the water in the part FGH begins to sink.

418. It was while repeating this experiment that Professor Magnus observed the water in GI lowering to the pipe ABC, and air was found to enter that pipe along with the stream. Since this was the case, it occurred to him, that a jet projected horizontally with considerable velocity against the orifice of a vessel full of water, would prevent that water from flowing out. In order to show this, he had a reservoir kept constantly full, from the bottom of which a right-angulartube gbg proceeded, gh being 7 feet long; the horizontal jet entered a tube de, communicating with the vessel A, 8 inches wide and 10 high. The tube de did not exceed 6 inches, f was distant from e by 6 inches, and the diameters of f and e were as 1:4. Under such conditions, the jet from f raised the surface of A to a height of 10 inches, without a single drop escaping from e; but when e had its diameter increased, a small portion of water flowed out from e ere the above height was attained. During the experiment a violent foaming took place within the tube de, which caused a vibratory motion in A.

He modified the above experiment by making the orifice of the vessel in the bottom, and projecting the jet vertically upwards, the diameters of tube and orifice being as 1:2. This diminution of the orifice was necessary from the fact, that the motion of the fluid exerts here a more disturbing influence than in the experiment with the horizontal tube.

419. After performing M. Savart's experiments of jets impinging on fixed planes, and against each other, the jet was directed against the interior surface of a hollow hemisphere of 24 mill. diameter. The jet issued from an orifice 3 mill. in diameter, and under a pressure of 2½ met.; the hemisphere was at right angles to the jet, and at a distance from the orifice of 0·5 met. The water, on leaving the hemisphere, directs itself to a point in the jet, but here it meets the coming stream, and so causes a peculiar foaming and streaming of the water, as represented in fig. 70.

When the jet was directed against the same hemisphere, obliquely situated with respect to the horizon, then the reflected water seeks to unite at a point below the line of jet, forming a figure similar to the preceding. The meeting point of the reflected water would be at a point above the jet were the hemisphere turned the other way.

420. These last two experiments are very important in the case where a jet, as we have seen, projected against an orifice prevents a flow from taking place; for the whole of the reflected water converges to a point in the jet, and, thus arrested, it must be thrown back so as to form a new surface. Hence, also, we can understand how a small jet can keep back a flow from a wide orifice, and also how a large surface cannot be formed, for it is broken up into a great number of little masses constituting foam.

421. In order to show this very clearly, Professor Magnus modified the experiment, by fixing on the horizontal tube de, fig. 72, a vertical part nm, the whole appearing like a T reversed (L).

The experiment was so arranged that k, the point where the jet met the water from A, lay between d and m; water was then poured in by the tube nm, but none of it escaped by e, for the whole was driven into A. The point k evidently depends on the pressure of A when f is constant; hence an increase of pressure throws k nearer to e.

He next inserted tightly a tube on n, passing under one side of the jet into a large empty bottle having a double head; a bent tube passed from the other head of this bottle into a jar, the open end of the former dipping into a coloured liquid or mercury. On the jet being set a-working, its action drew out the air from the new tubes by nm, and accordingly the coloured fluid or mercury rose from its vessel.

422. When the jet meets the water in de, a foaming takes place, and the air inclosed in the foam enters the vessel A. That this is the case may very easily be seen when A is a glass vessel; bubbles of air will then stream up to the surface-opening of A. Savart had previously noticed this same thing, but thought it of no consequence; Prof. Magnus, on the contrary, thinks it highly important. Almost every individual has noticed, of a morning, bubbles of air clinging to the sides of the tumbler; how have these obtained entrance? Prof. Magnus thinks he can account for it, and the explanation of it brings him a second time to refute Venturi's assertion that light bodies are carried along with the jet. For if this were the case, then air would enter the vessel A by the friction of the air and jet; but this is impossible, for the air cannot be put in circulation by a jet spouting perpendicularly from a vessel by a long tube, its mouth being distant from the surface of the water impinged on by 1 mill., as was shown by Prof. Magnus.

But the phenomenon appears to be caused in the following manner: When a jet strikes the water a hollow is immediately formed on the surface; this closes in very quickly, as soon as the least motion is imparted to the surface, and while closing up, so much air is carried downwards. When the jet is continuous, these hollows, which are distinctly visible, are so speedily formed and so quickly closed up, that it is almost impossible to detect how the air gets introduced.

Prof. Magnus, however, found that when a jet falls on a tranquil surface placed at a small distance from the orifice, and before it has attained its greatest contraction, a considerable concavity may be seen around the jet without any air entering; let the surface now be slightly disturbed by a few drops falling upon it from a height of a few inches, and at a small distance from the jet. The drops do not of themselves carry any air downwards; but whenever they meet the surface, a peculiar sound proceeds from the part of the surface impinged on by the jet, and at the same instant air-bubbles form which carry air downwards.

423. Prof. Magnus now directed his attention to the action which a jet has on a body placed in a fluid, or to determine experimentally if the resistance varies with the distance from the orifice. In order to render the results as complete as possible, experiments were first made in air, to ascertain how far it is necessary that the plates used should be flat; and to render the results independent of the gravity of the water, a horizontal stream was applied. The resistance was in each instance measured by a kind of balance to which the plate was attached. Smooth flat plates and hollow hemispheres were preferred. The diameter of the plates varied from 100 to 200 mill. The plates were placed in a vessel of water, first vertically at a certain distance from the orifice of a horizontal jet, and second, horizontally or parallel with the surface, and subject to a vertical jet. The tables, as under, will show the results of the several experiments.

| HORIZONTAL JET | VERTICAL JET | |---------------|-------------| | Orifice | Vertical pressure | Diameter of plate | Distance of plate from orifice | Weight in grammes necessary to bring the plate to its place | |---------------|------------------|-------------------|-------------------------------|----------------------------------------------------------| | In thin plate, diameter 3 mill. | 2-145 | 100 | 20 | 20-0 | No. 1 | 20-0 | 20-0 | 20-0 | | | | | 50 | 21-0 | No. 2 | 21-0 | 20-75 | | | | | 100 | 21-5 | No. 3 | 21-5 | 21-5 | | | | | 150 | 21-5 | | 21-5 | 21-5 | | | | | 200 | 21-5 | | 20-5 | 20-5 | | In thin plate, diameter 3 mill. | 2-145 | 150 | 20 | 20-0 | No. 4 | 20-0 | 18-0 | | | | | 50 | 21-0 | No. 5 | 21-0 | 19-0 | | | | | 100 | 22-0 | | 20-0 | 20-0 | | | | | 150 | 23-0 | | 23-5 | 20-0 | | | | | 200 | 23-0 | | 23-5 | 21-0 | | | | | 250 | 23-0 | | 23-0 | 21-0 | | | | | 300 | 22-5 | | 22-5 | 20-5 | | In brass 1 mill. thick, diameter 3 mill. | 2-229 | 200 | 20 | 16-0 | No. 7 | 16-2 | | | | | 50 | 16-7 | | 16-7 | | | | | 100 | 18-0 | | 18-0 | | | | | 150 | 18-0 | | 18-0 | | | | | 200 | 18-0 | | 18-0 | | | | | 250 | 17-5 | | 17-7 | | Small glass tube, 10 mill. long, diameter 3 mill. nearly. | 2-229 | 150 | 20 | 21-0 | No. 9 | 21-0 | | | | | 50 | 21-6 | | 21-6 | | | | | 100 | 23-2 | | 23-2 | | | | | 150 | 23-3 | | 23-3 | | | | | 200 | 23-3 | | 23-3 | | | | | 250 | 23-2 | | 23-2 | | Small glass tube, 20 mill. long, diameter a little under 3 mill. | 2-229 | 150 | 20 | 14-3 | No. 10 | 14-3 | | | | | 50 | 14-9 | | 14-9 | | | | | 100 | 15-2 | | 15-2 | | | | | 150 | 15-4 | | 15-4 | | | | | 200 | 15-2 | | 15-2 |

424. In determining the above results, it was found that, the distance from the orifice remaining constant, the force requisite to produce an equilibrium increased with the magnitude of the plate up to a certain limit, beyond which the force was constant. In the case of the horizontal jet, the pressure on the plate, up to a length of 150 mill., increased with the distance from the orifice. None of the experiments showed a decrease within the distance of 100 mill. But the increase was not always equal, and this was owing to the imperfect nature of the apparatus. A similar increase was shown in the case of the vertical jet, both when it struck the centre of the plate vertically, and when at an angle of 10°.

425. The last thing that Prof. Magnus endeavoured to resolve in these experiments, was the important one of how a jet of water mixes with the water of a vessel into which it is projected. He first of all introduced into a large tank, nearly full of water, various bodies, as *Semen lycopodii* and milk; but on the jet playing into the vessel, the motion of the fluid was unsteady and violent, so that nothing could be determined. A very curious phenomenon, however, was observed; no sooner had the jet begun to play in the vessel, at a distance of 2 inches under the surface, than the fluid surface immediately over the jet was more or less depressed, according to the greater or less power of the jet. Sometimes the surface was a little above, sometimes a little below the jet; at the jet the surface was very uneven and wavy. The backward and forward motion of the water under the action of the jet, showed that a considerable portion of the motionless water must be carried forward by the jet; this causes a whirling motion in the water, which, owing to its opacity, prevents the nature of the action from being known.

The next process which he adopted so as to arrive at a knowledge of what he wanted, was to cause water, free from salt and hydrochloric acid, to spout into a vessel containing water, into which was dissolved about 1 per cent. of salt. The exact amount of salt in the mass was accurately determined by a solution of nitrate of silver. In order then to take water from the jet streaming into and below the surface of this solution, he took a long bent tube such that one extremity was in the water and opposite the jet, while the other extremity rose above the surface of water, and emptied into a tumbler any water that flowed through it. If now a jet spout into the vessel, and one extremity of the bent tube be applied at certain distances from, and in opposition to the jet, so much fluid will enter the tube and pass into the tumbler; by the proper test the amount of salt in the tumbler may easily be detected. When the distance between the tube and the orifice of the jet was considerable, the fluid received by the tube into the tumbler was the same as that in the vessel. Three separate portions were taken in the glass and examined, and when the second and third were of the same composition, the result was considered as correct. As the jet continued to spout in, the water in the vessel was more and more diluted, but before each experiment the fluid was always tested.

It was of the utmost importance to know whether the salt solution penetrated the middle of the flowing jet; and to this end the tube was accurately fixed in direction with the jet while the vessel was empty. The following table will show the results of the several experiments.

### The Tube Point in the middle of the Jet

| Number of experiments | Diameter of orifice through which fluid entered, in mill. | Quantity of salt contained in vessel before experiment, per cent. | Quantity of salt contained in fluid obtained through tube, per cent. | The fluid obtained through tube consisted of | |-----------------------|----------------------------------------------------------|---------------------------------------------------------------|---------------------------------------------------------------|---------------------------------------------| | 1 | 3 | 10 | 0.92 | 0.50 | 54.4 | 45.6 | | 2 | 3 | 20 | 0.50 | 0.54 | 60.0 | 40.0 | | 3 | 3 | 20 | 0.78 | 0.46 | 59.0 | 41.0 | | 4 | 5 | 10 | 0.89 | 0.03 | 34.6 | 65.4 | | 5 | 5 | 20 | 0.93 | 0.14 | 15.1 | 84.9 | | 6 | 5 | 20 | 0.88 | 0.14 | 14.8 | 85.2 | | 7 | 5 | 30 | 0.93 | 0.27 | 29.0 | 71.0 | | 8 | 5 | 30 | 0.90 | 0.27 | 30.0 | 70.0 | | 9 | 5 | 30 | 0.88 | 0.23 | 25.2 | 74.8 | | 10 | 5 | 30 | 0.82 | 0.19 | 23.2 | 76.8 | | 11 | 5 | 30 | 0.99 | 0.22 | 22.0 | 78.0 | | 12 | 5 | 50 | 0.98 | 0.44 | 44.9 | 55.1 |

When the point of the tube was brought close up to the orifice, the pure water of the jet flowed by the tube into the tumbler, and the test showed not the smallest trace of salt. The results, however, were different when the tube point was slightly inclined to the direction of the jet.

These results are shown in the following table:

### The Tube Point inclined to the Jet

| Number of experiments | Distance from orifice, in mill. | The tube was situated | Quantity of salt contained before experiment, per cent. | Quantity of salt contained through tube, per cent. | The fluid obtained through tube consisted of | |-----------------------|---------------------------------|------------------------|--------------------------------------------------------|---------------------------------------------------|---------------------------------------------| | 13 | 10 | In the middle | 0.89 | 0.63 | 3.4 | 96.6 | | 13 | 10 | 1'75 sideways | 1.68 | 0.93 | 21.3 | 78.7 | | 13 | 10 | 3' sideways | 0.94 | 0.42 | 44.7 | 55.3 | | 6 | 20 | In the middle | 0.98 | 0.16 | 16.4 | 83.6 | | 6 | 20 | 1'75 sideways | 1.66 | 0.40 | 37.7 | 62.3 | | 6 | 20 | 2'5 sideways | 0.99 | 0.45 | 45.5 | 54.5 | | 11 | 30 | In the middle | 0.99 | 0.22 | 22.0 | 78.0 | | 11 | 30 | 1'75 sideways | 0.97 | 0.46 | 47.4 | 52.6 | | 11 | 30 | 2'5 sideways | 0.95 | 0.56 | 58.3 | 41.7 |

This table lets us see that the amount of salt in the drawn off fluid, increases considerably as the deviation from the direction of the jet increases.

426. In order to explain the preceding phenomena, suppose that a jet enters an indefinite mass of fluid, by an orifice at a depth below the fluid surface, then the velocity of the jet will decrease as its distance from the orifice increases, and its dimensions will also increase. Take the volume which will enter the mass in a unit of time; then this volume expands in a direction perpendicular to the axis of the jet, and its dimension in the direction of the jet must diminish. If the velocity decrease in such a manner, that both velocity and thickness are inversely as the cross section during the same time, then the same mass will pass through all planes perpendicular to the axis of the jet in the same space of time; but, in such a case, the force imparting motion to the water must decrease as the velocity decreases. The moving force, however, cannot be destroyed, since the resistance to motion from the sides of the indefinite fluid mass is nothing, and the surface is supposed to continue horizontal. The jet, doubtless, is resisted by the particles of the indefinite mass, but since that is due to the inertia of the mobile particles, the moving force in the direction of the jet is unaltered; and hence, as soon as the motion has attained a degree of permanency, the same amount of force is exerted during the unit of time in all planes perpendicular to the axis of the jet. If, when the jet widened, the volume underwent a change of shape, such that its cross section became greater while the velocity remained unchanged, the time occupied by the mass in passing through the different cross sections would be less and less as the width increased. Were this the case, the different layers of the jet must either separate, or their density must change. If care be taken that the same mass alone expands, and no addition be made to it, then a decrease of pressure, without a change of density taking place, will be the result. But again, since the mass moves in a fluid of the same kind, an increase of volume will take place from the lateral communication of the jet and surrounding fluid. This increase of mass diminishes the velocity; but the moving force being constant, a greater volume of fluid will pass through a distant section in a certain time, than through one nearer the orifice.

We shall see presently that the pressure of a moving fluid is less than if it were still; and therefore to say that no portion of the moving force is lost when the jet is in action might seem somewhat inconsistent, since part of this force... would seem necessary to balance the difference of pressure in liquids. But there is no inconsistency in this, for the diminution of pressure is caused by the forward portions of the fluid passing through a cross section of the jet quicker than those which follow. If, then, the difference of pressure lessens the velocity of the original mass, it also sets the fluid at each side in motion, and in this way is the loss of force compensated.

The consequences which follow from the above are—

1. That a jet streaming into a fluid, similar to itself, sends more water, in the same time, through a cross section situated at a distance from the orifice, than through one nearer the same.

2. That in consequence of this, the pressure of the fluid when in motion is less than when in a state of rest.

These two propositions will explain the preceding phenomena. In the experiment stated in 426, more water passes through a cross section of the current EG, than passes in the same time through the tube ACD; wherefore a portion of the water must be carried off from the vessel CDG. So also (fig. 68), more water passes through the communicating tube ABC, than through the narrow tube α; and as no lateral motion takes place, a portion of the water in HF will pass along ABC, and its level falls. If the surface in HF should sink, while the overplus of water in HE flows out by K, or if there be a constant level in HE, then the jet is affected by the pressure due to the difference of level, and the moving force is diminished, so that it is less at a cross section of ABC at A, than in the jet tube α. Therefore, the volume of fluid which passes in a unit of time through A will decrease, and the fluid surface in HF will continue to sink, until the volume of fluid passing through a section of A in a unit of time, be equal to a volume which, during the same time, passes through A. The difference of level will enable us to determine the pressure.

If, on the fluid surface of HF reaching the lower face of ABC, it happen that the above condition of equality of volumes be not fulfilled, then the water in the tube is less pressed in the direction from A to B than from B to A, although the statical pressure in the latter direction is that due to the atmosphere, while in the former it is the atmospheric pressure plus the weight of a column the height of which is that due to the difference of levels. This excess of pressure at B causes the air to enter the tube in bubbles.

427. The same is the explanation of the phenomena attendant on the experiment (fig. 69) where a thin jet prevented the flow of water from an orifice in a vessel: the air entered the water through that tube, and it continued to do so till the fluid in the vessel attained such a height, that the pressures at each end of the tube were equal. If the inner pressure be in excess, the water flows out by the orifice.

428. In conducting these experiments, he found that the pressure of a moving fluid was less than that of a fluid at rest. Thus, when the fluid is stopped by the plate, the water, being resisted in its progress, now moves along parallel to the plate, which, when of a proper magnitude, must diminish the pressure against the plate. Therefore, necessarily, since fluids press equally in all directions (art. 48), the pressure from behind the plate must be greater than on that side facing the jet. This difference will increase if we increase the velocity of the jet and the magnitude of the plate. If the plate be brought nearer the orifice, the velocity of the jet will be greater upon the plate, for then the vein has not had time to widen, and consequently the space through which the water has to pass parallel to the plate is increased. Hence, the nearer that the plate is to the orifice, the greater is the difference of pressure before and behind, and the less will be the force required to bring the plate back to its position of equilibrium. This result continues till the difference of pressure vanishes. This fact may be shown experimentally, by having one plate parallel to another, both vertical, one fixed and having an orifice, the other free and connected with a balance; then when a jet is sent through the former, the difference of pressure on the latter will force it nearer to the fixed plate. The same thing will take place when a column of air is made to act on a plate similarly circumstanced as the above.

429. The whirling motion stated in article 425 is due to the lateral motion imparted to the water by a strong jet. If the orifice be not too small and expanded vertically, and if the water have a sufficient velocity, funnel-shaped cavities are formed; for the water within these small whirlpools continually streams towards their outer rim, and so the pressure at the centre is diminished. Many other phenomena are dependent on this lateral motion.

7. Experiments of M. V. Regnault.

430. In the Mémoires de l'Institut for 1847, M. Regnault gives an account of his experiments, which were undertaken for the purpose of determining the principal laws and numerical data which enter into the calculation of steam-engines. The nature of his investigations necessarily led him to experiment on the compressibility of water and mercury, and his deductions on this subject may be considered as very accurate. The process which Oersted, Colladon and Sturm, and Aimé adopted was very simple, and generally speaking, may be described as follows:—A liquid is placed in a kind of thermometer or vessel of considerable capacity surmounted by a graduated capillary tube, open at the top, the divisions being for finding the ratio of the volume between any two divisions, and that of the vessel. This apparatus or piezometer, is placed in an experimenting glass vessel, with strong walls, and furnished with a metal covering, which communicates on one side with a forcing-pump, and on the other with a manometer. The latter strong vessel is filled with water, and as soon as the pump begins to work, a pressure, more or less great, is exercised on the sides of the vessel; at the same time the internal and external walls of the piezometer are subjected to compression, and the liquid acted on is diminished in volume. The level of the liquid in the graduated tube now falls, it may be, through several divisions; but this will be a means of calculating the diminution of volume that the liquid, subject to pressure, as measured by the manometer, has experienced. It is only, however, the apparent, not the absolute compressibility, owing to the change of dimension of the piezometer during the operation. Oersted supposed that no change took place.

431. M. Regnault's apparatus consisted also of a piezometer, but so disposed that he obtained at once the compressibility of the liquid, and that of the envelope containing it. He employed air to exercise the compression, because it presents greater advantages than when water is pumped in; the pressures, e.g., are always constant, and they may be measured with great precision. The piezometers he used were most carefully constructed, being first made into hemispheres, and then soldered together. The following table will show us the mean results obtained:

| Envelope or Piezometer | Pressure in atmospheres varying from | Compressibility of water | |------------------------|-----------------------------------|-------------------------| | Spherical, copper | 2-8033 to 7-8220 | Apparent: 0-000046392 | | | | Theoretical: 0-000047709 | | Spherical, brass | 1-5842 to 9-1206 | Apparent: 0-000046847 | | | | Theoretical: 0-000048288 | | Cylindrical, glass | 2-5387 to 10-3528 | Apparent: 0-000044394 | | | | Theoretical: 0-000049077 | | Cylindrical, glass | 3-3338 to 9-5408 | Apparent: 0-000001234 | | | | Theoretical: 0-000003517 |

The tabulated results of M. Regnault, of which the first on the table is the mean, show that as the pressure in- creased, the compressibility of the water sensibly diminished, and that of the envelope was irregular; this was owing to the high malleable and low elastic power of the copper.

The results of those tables, of which the second and the third are the mean, did not show this diminution, owing, perhaps, to the high elastic power of the brass and glass, and the low malleable nature of the former. Hence, when water is subjected to a pressure of 15 lbs. on the square inch, or one atmosphere, it is diminished by 48 volumes in one million of the same; i.e., if there be a million cubic inches of water, then if the mass be subjected to a pressure of 1, 2, 3, 4, &c., atmospheres, the diminution of its volume will be $1 \times 48$, $2 \times 48$, $3 \times 48$, $4 \times 48$, &c., cubic inches.

When mercury was subjected to pressure, the tabulated results, of which the fourth is the mean, showed that the apparent differed much from the theoretical compressibility, owing probably to the difficulty of the operation. The liquid, however, is regarded as compressible to the extent of 35 volumes in every 10 million, for each compressing atmosphere.

CHAPTER III.—ON THE RESISTANCE OF FLUIDS.

432. The celebrated Coulomb has very successfully employed the principle of torsion, to determine the cohesion of fluids, and the laws of their resistance in very slow motions. When a body is struck by a fluid with a velocity exceeding eight or nine inches per second, the resistance has been found proportional to the square of the velocity, whether the body in motion strikes the fluid at rest, or the body is struck by the moving fluid. But when the velocity is so slow as not to exceed four-tenths of an inch in a second, the resistance is represented by two terms, one of which is proportional to the simple velocity, and the other to the square of the velocity. The first of these sources of resistance arises from the cohesion of the fluid particles which separate from one another, the number of particles thus separated being proportional to the velocity of the body. The other cause of resistance is the inertia of the particles, which, when struck by the fluid, acquire a certain degree of velocity proportional to the velocity of the body; and as the number of these particles is also proportional to that velocity, the resistance generated by their inertia must be proportional to the square of the velocity.

When the body in motion, therefore, meets the fluid at rest, Sir Isaac Newton, Daniel Bernoulli, and M. Gravesende maintained that the formula which represents the resistance of fluids consists of two terms, one of which is as the square of the velocity, and the other constant. The experiments of Coulomb, however, incontestably prove that the pressure which the moving body in this case sustains, is represented by two terms, one proportional to the simple velocity, and the other to its square, and that if there is a constant quantity, it is such as escapes detection.

433. In order to apply the principle of torsion to the resistance of fluids, M. Coulomb made use of the apparatus represented in fig. 73. On the horizontal arm LK, which may be supported by a vertical stand, is fixed the small circle fe, perforated in the centre, so as to admit the cylindrical pin ba. Into a slit in the extremity of this pin is fastened, by means of a screw, the brass wire ag, whose force of torsion is to be compared with the resistance of the fluid; and its lower extremity is fixed in the same way into a cylinder of copper gd, whose diameter is about four-tenths of an inch. The cylinder gd is perpendicular to the disk DS, whose circumference is divided into 480 equal parts. When this horizontal disk is at rest, which happens when the torsion of the brass wire is nothing, the fixed index RS is placed upon the point O, the zero of the circular scale. The small rule Rm may be elevated or depressed at pleasure round its axis n, and the stand CH which supports it may be brought into any position round the horizontal disk. The lower extremity of the cylinder gd is immersed about two inches in the vessel of water MNOP, and to the extremity d is attached the planes, or the bodies whose resistance is to be determined when they oscillate in the fluid by the torsion of the brass wire. In order to produce these oscillations, the disk DS, supported by both hands, must be turned gently round to a certain distance from the index, without deranging the vertical position of the suspended wire. The disk is then left to itself; the force of torsion causes it to oscillate, and the successive diminutions of these oscillations are carefully observed.

434. The method employed by Coulomb, in reducing his experiments, is similar to that adopted by Newton and other mathematicians, when they wished to determine the resistance of fluids, from the successive diminutions of the oscillations of a pendulum moving in a resisting medium; but is much better fitted for detecting the small quantities which are to be estimated in such researches. When the pendulum is employed, the specific gravity of the body, relative to that of the fluid, must be determined; and the least error in this point leads to very uncertain results. When the pendulum is in different points of the arc in which it oscillates, the wire or pendulum rod is plunged more or less in the fluid; and the alterations which may result from this are frequently more considerable than the small quantities which are the object of research. It is only in small oscillations too, that the force which brings the pendulum from the vertical, is proportional to the angle which the pendulum rod, in different positions, forms with this vertical line; a condition which is necessary before the formula can be applied. But small oscillations are attended with great disadvantages; and their successive diminutions cannot be determined but by quantities which it is difficult to estimate exactly, and which are changed by the smallest motion either of the fluid in the vessel, or of the air in the chamber. In small velocities, the pendulum rod experiences a greater resistance at the point of flotation than at any other part. This resistance too, is very changeable; for the water rises from its level along the pendulum rod to greater or less heights, according to the velocity of the pendulum.

435. These and other inconveniences which might be mentioned, are so inseparable from the use of the pendulum, that Newton and Bernoulli have not been able to determine the laws of the resistance of fluids in very slow motions. When the resistance of fluids is compared with the force of torsion, these disadvantages do not exist. The body is in this case entirely immersed in the fluid; and as every point of its surface oscillates in a horizontal plane, the relation between the densities of the fluid and the oscillating body has no influence whatever on the moving force. One or two circles of amplitude may be given to the oscillations; and their duration may be increased at pleasure, either by diminishing the diameter of the wire, or increasing its length; or, which may be more convenient, by augmenting the momentum of the horizontal disk. Coulomb, however, found that when each oscillation was so long as to continue about 100 seconds, the least motion of the fluid, or the tremor occasioned by the passing of a carriage, produced a sensible alteration on the results. The oscillations best fitted for experiments of this kind continued from 20 to 30 seconds, and the amplitude of those that gave the most regular results was comprehended between 480 degrees, the entire division of the disk, and 8 or 10 divisions reckoned from the zero of the scale. From these observations, it will be readily seen, that it is only in very slow motions that an oscillating body can be employed for determining the resistance of fluids. In small oscillations, or in quick circular motions, the fluid struck by the body is continually in motion; and when the oscillating body returns to its former position, its velocity is either increased or retarded by the motion communicated to the fluid, and not extinguished.

436. In the first set of experiments made by Coulomb, he attached to the lower extremity of the cylinder \( gd \) a circular plate of white iron, about 195 mill. in diameter, and made it move so slowly that the part of the resistance proportional to the square of the velocity wholly disappeared. For if, in any particular case, the portion of the resistance proportional to the simple velocity should be equal to the portion that is proportional to the square of the velocity when the body has a velocity of one-tenth of an inch per second, then, when the velocity is 100 tenths of an inch per second, the part proportional to the square of the velocity will be a hundred times greater than that proportional to the simple velocity; but if the velocity is only the hundredth part of the tenth of an inch per second, then the part proportional to the simple velocity will be 100 times greater than the part proportional to the square of the velocity.

437. When the oscillations of the white-iron plate were so slow, that the part of the resistance which varies with the second power of the velocity was greatly inferior to the other part, he found, from a variety of experiments, that the resistance which diminished the oscillations of the horizontal plate was uniformly proportional to the simple velocity; and that the other part of the resistance, which follows the ratio of the square of the velocity, produced no sensible change upon the motion of the white-iron disk. He also found, in conformity with theory, that the momenta of resistance in different circular plates moving round their centre in a fluid, are as the fourth power of the diameters of these circles; and that, when a circle of 195 mill. (7.677 English inches) in diameter, moved round its centre in water, so that its circumference had a velocity of 140 mill. (5.512 English inches) per second, the momentum of resistance which the fluid opposed to its circular motion was equal to one-tenth of a gramme (1.544 English troy grains) placed at the end of a lever 143 mill. (5.63 English inches) in length.

438. M. Coulomb repeated the same experiments in a vessel of clarified oil, at the temperature of 16 degrees of Reaumur. He found, as before, that the momenta of the resistance of different circular disks, moving round their centre in the plane of their superficies, were as the fourth power of their diameters; and that the difficulty with which the same horizontal plate, moving with the same velocity, separated the particles of oil, was to the difficulty with which it separated the particles of water, as 175 to 1, which is therefore the ratio that the mutual cohesion of the particles of oil has to the mutual cohesion of the particles of water.

439. In order to ascertain whether or not the resistance of a body moving in a fluid was influenced by the nature of its surface, M. Coulomb anointed the surface of the white-iron plate with tallow, and wiped it partly away, so that the thickness of the plate might not be sensibly increased. The plate was then made to oscillate in water, and the oscillations were found to diminish in the same manner as before the application of the unguent. Over the surface of the tallow upon the plate, he afterwards scattered, by means of a sieve, a quantity of coarse sand which adhered to the greasy surface; but when the plate, thus prepared, was caused to oscillate, the augmentation of resistance was so small that it could scarcely be appreciated. We may therefore conclude, that the part of the resistance which is proportional to the simple velocity, is owing to the mutual adhesion of the particles of the fluid, and not to the adhesion of these particles to the surface of the body.

440. If the part of the resistance varying with the simple velocity were increased when the white-iron plate was immersed at greater depths in the water, we might suppose it to be owing to the friction of the water on the horizontal surface, which, like the friction of solid bodies, should be proportional to the superincumbent pressure. In order to settle this point, M. Coulomb made the white-iron plate oscillate at the depth of two centimetres (0.787 English inches), and also at the depth of 50 centimetres (19.6855 English inches), and found no difference in the resistance; but as the surface of the water was loaded with the whole weight of the atmosphere, and as an additional load of 50 centimetres of water could scarcely produce a perceptible augmentation of the resistance, M. Coulomb employed another method of deciding the question. Having placed a vessel full of water under the receiver of an air-pump, the receiver being furnished with a rod and collar of leather at its top, he fixed to the hook, at the end of the rod, a harpsichord wire, number 7 in commerce, and suspended to it a cylinder of copper, like \( gd \), fig. 89, which plunged in the water of the vessel, and under this cylinder he fixed a circular plane, whose diameter was 101 millimetres (3.976 English inches). When the oscillations were finished, and consequently the force of torsion nothing, the zero of torsion was marked by the aid of an index fixed to the cylinder. The rod was then made to turn quickly round through a complete circle, which gave to the wire a complete circle of torsion, and the successive diminutions of the oscillations were carefully observed. The diminution for a complete circle of torsion was found to be nearly a fourth part of the circle for the first oscillation, but always the same whether the experiment was made in a vacuum or in the atmosphere. A small pallet 50 millimetres long (1.969 English inches) and 10 millimetres broad (0.3937 English inches), which struck the water perpendicular to its plane, furnished a similar result. We may therefore conclude, that when a submerged body moves in a fluid, the pressure which it sustains, measured by the altitude of the superior fluid, does not perceptibly increase the resistance; and, consequently, that the part of this resistance proportional to the simple velocity, can in no respect be compared with the friction of solid bodies, which is always proportional to the pressure.

441. The next object of M. Coulomb was to ascertain the resistance experienced by cylinders that moved very slowly, and perpendicular to their axis; but as the particles of fluid struck by the cylinder necessarily partook of its motion, it was impossible to neglect the part of the resistance proportional to the square of the velocity, and there- fore he was obliged to perform the experiments in such a manner that both parts of the resistance might be computed. The three cylinders which he employed were 249 mill. (98031 English inches) long. The first cylinder was 0-87 mill. (06342 English inches, or \( \frac{5}{8} \)th of an inch) in circumference, the second 11-2 mill. (04409 English inches), and the third 21-1 mill. (83070 English inches). They were fixed by their middle under the cylindrical piece \( Dq \), so as to form two horizontal radii, whose length was 124-5 mill. (49015 English inches) or half the length of each cylinder. After making the necessary experiments and computations, he found that the part of the resistance proportional to the simple velocity, which, to avoid circumlocution, we shall call \( r \), did not vary with the circumferences of the cylinders. The circumferences of the first and third cylinders were to one another as 24 : 1, whereas the resistances were in the ratio of 3 : 1. The same conclusion was deduced by comparing the experiments made with the first and second cylinder.

442. In order to explain these results, M. Coulomb very justly supposes, that, in consequence of the mutual adhesion of the particles of water, the motion of the cylinder is communicated to the particles at a small distance from it. The particles which touch the cylinder have the same velocity as the cylinder, those at a greater distance have a less velocity, and at the distance of about one-tenth of an inch the velocity ceases entirely, so that it is only at that distance from the cylinder that the mutual adhesion of the fluid molecules ceases to influence the resistance. The resistance \( r \), therefore, should not be proportional to the circumference of the real cylinder, but to the circumference of a cylinder whose radius is greater than the real cylinder by one-tenth of an inch. It consequently becomes a matter of importance to determine with accuracy the quantity which must be added to the real cylinder in order to have the radius of the cylinder to which the resistance \( r \) is proportional, and from which it must be computed. Coulomb found the quantity by which the radius should be increased to be 1-5 mill. (\( \frac{5}{8} \)ths of an English inch), so that the diameter of the augmented cylinder will exceed the diameter of the real cylinder by double that quantity, or \( \frac{1}{4} \)ths of an inch.

443. The part of the resistance varying with the square of the velocity, or that arising from the inertia of the fluid, which we shall call \( R \), was likewise not proportional to the circumferences of the cylinder; but the augmentation of the radii amounts in this case only to \( \frac{1}{4} \)ths of an inch, which is only one-fifth of the augmentation necessary for finding the resistance \( r \). The reason of this difference is obvious; all the particles of the fluid when they are separated from each other oppose the same resistance, whatever be their velocity; consequently, as the value of \( r \) depends only on the adhesion of the particles, the resistances due to this adhesion will reach to the distance from the cylinder where the velocity of the particles is 0. In comparing the different values of \( R \), the part of the resistance which varies as the square of the velocity, all the particles are supposed to have a velocity equal to that of the cylinder; but as it is only the particles which touch the cylinder that have this velocity, it follows that the augmentation of the diameter necessary for finding \( R \) must be less than the augmentation necessary for finding \( r \).

444. In determining experimentally the part of the momentum of resistance proportional to the velocity, by two cylinders of the same diameter, but of different lengths, M. Coulomb found that this momentum was proportional to the third power of their lengths. The same result may be deduced from theory; for supposing each cylinder divided into any number of parts, the length of each part will be proportional to the whole length. The velocity of the corresponding parts will be as these lengths, and also as the distance of the same parts from the centre of rotation. The theory likewise proves that the momentum of resistance depending on the square of the velocity, in two cylinders of the same diameter but of different lengths, is proportional to the fourth power of the length of the cylinder.

445. When the cylinder, 98031 inches in length, and 04409 inches in circumference, was made to oscillate in the fluid with a velocity of 5-31 inches per second, the part of the resistance \( r \) was equal to 58 milligrammes, or \( \frac{943}{1000} \) Troy grains. And when the velocity was 0-3937 inches per second, the resistance \( r \) was 000414 grammes, or \( \frac{638}{1000} \) Troy grains.

446. The preceding experiments were also made in the oil formerly mentioned; and it likewise appeared, from their results, that the mutual adhesion of the particles of oil was to the mutual adhesion of the particles of water as 17 to 1. But though this be the case, M. Coulomb discovered that the quantity by which the radii of the cylinder must be augmented in order to have the resistance \( r \), is the very same as when the cylinder oscillated in water. This result was very unexpected, as the greater adhesion between the particles of oil might have led us to anticipate a much greater augmentation. When the cylinders oscillated both in oil and water with the same velocity, the part of the resistance \( R \) produced by the inertia of the fluid particles which the cylinder put in motion, was almost the same in both. As this part of the resistance depends on the quantity of particles put in motion, and not on their adhesion, the resistances due to the inertia of the particles will be in different fluids as their densities.

447. The subject of the resistance of fluids has been treated by Dr Hutton of Woolwich. His experiments were made in air with bodies of various forms, moving with different velocities, and inclined at various angles to the direction of their motion, and the following conclusions were deduced from them:

1. That the resistance is nearly proportional to the surfaces, a small increase taking place when the surfaces and the velocities are great. 2. The resistance to the same surface moving with different velocities, is nearly as the square of the velocity; but it appears that the exponent increases with the velocity. 3. The round and sharp ends of solids sustain a greater resistance than the flat ends of the same diameter. 4. The resistance to the base of the hemisphere is to the resistance on the convex side, or the whole sphere, as \( \frac{2}{3} \) to 1, instead of \( \frac{2}{3} \) to 1, as given by theory. 5. The resistance on the base of the cone is to the resistance on the vertex nearly as \( \frac{2}{3} \) to 1; and in the same ratio is radius to the sine of half the angle at the vertex. Hence in this case the resistance is directly as the sine of the angle of incidence, the transverse section being the same. 6. The resistance of the base of a hemisphere, the base of a cone, and the base of a cylinder, are all different, though these bases be exactly equal and similar.

448. The experiments of Mr Vince were made with bodies at a considerable depth below the surface of water; and he determined the resistance which they experienced, both when they moved in the fluid at rest, and when they received the impulse of the moving fluid. In the experiments contained in the following table, the body moved in the fluid with a velocity of 0-96 feet in a second. The angles at which the planes struck the fluid are contained in the first column. The second column shows the resistance by experiment in the direction of their motion in Troy ounces. The third column exhibits the resistance by theory, the perpendicular distance being supposed the same as by experiment. The fourth column shows the power of the sine of the angle to which the resistance is proportional, and was computed in the following manner:—Let \( s \) be the sine of the angle, radius being 1, and \( r \) the resistance at that angle. Suppose \( r \) to vary as \( s^n \), then we have On the Resistance \( r^n : s^n = 0.2321 : r \); hence \( r^n = \frac{r^{n+1}}{0.2321} \) and therefore

\[ m = \frac{\log_r - \log_s}{\log_r - \log_s}, \quad \text{and by substituting their corresponding values, instead of } r \text{ and } s \text{ we shall have the values of } m \text{ or the numbers in the fourth column.} \]

**Resistance of a Plane Surface moving in a Fluid, and placed at different angles to the path of its motion.**

| Angle of Inclination | Resistance by experiment | Resistance by theory | Power of the sine of the angle to which the resistance is proportional | |---------------------|--------------------------|----------------------|---------------------------------------------------------------| | Degrees | Troy ounces | Troy ounces | Experiments | | 10 | 0.0112 | 0.0012 | 173 | | 30 | 0.0769 | 0.0290 | 154 | | 50 | 0.1552 | 0.1043 | 161 | | 70 | 0.2125 | 0.1926 | 142 | | 90 | 0.2321 | 0.2321 | ... |

449. According to the theory the resistance should vary as the cube of the sine, whereas from an angle of 90° it decreases in a less ratio, but not as any constant power, nor as any function of the sine and cosine. Hence the actual resistance always exceeds that which is deduced from theory, assuming the perpendicular resistance to be the same. The cause of this difference is partly owing to our theory neglecting that part of the force which after resolution acts parallel to the plane, but which, according to experiments, is really a part of the force which acts upon the plane.

450. Mr Vince made also a number of experiments on the resistance of hemispheres, globes, and cylinders, which moved with a velocity of 0.542 feet per second. He found that the resistance to the spherical side of a hemisphere was to the resistance on its base as 0.034 is to 0.08339; that the resistance of the flat side of a hemisphere was to the resistance of a cylinder of the same diameter, and moving with the same velocity, as 0.08339 is to 0.07998; and that the resistance to a complete globe is to the resistance of a cylinder of the same diameter, and with the same velocity, as 1 : 2.23.

When the plane was struck by the moving fluid, Dr Vince found that the resistance was one-fifth greater than when the plane moved in the fluid. In both these cases the actual effect on the plane must be the same, and therefore the difference in the resistance can arise only from the action of the fluid behind the body in the former case.

**CHAPTER IV.—ON THE OSCILLATION OF FLUIDS, AND THE UNDULATION OF WAVES.**

451. **Prop. I.**—The oscillations of water in a siphon, consisting of two vertical branches and a horizontal one, are isochronous, and have the same duration as the oscillations of a pendulum whose length is equal to half the length of the oscillating column of water.

Into the tube MNOP (fig. 74), having its internal diameter everywhere the same, introduce a quantity of water. When the water is in equilibrium, the two surfaces AB, CD will be in the same horizontal line AD. If this equilibrium be disturbed by making the siphon oscillate round the point y, the water will rise and fall alternately in the vertical branches after the siphon is at rest. Suppose the water to rise to EF in the branch MO, it will evidently fall to GH in the other branch, so that CG is equal to AE. Then it is evident, that the force which makes the water oscillate, is the weight of the column EFKL, which is double the column EABF; and that this force is to the whole weight of the water, as 2 AE is to AOPD. Now, let P (fig. 75) be a pendulum whose length is equal to half the length of the oscillating column AOPD, and which describes to the lowest point S arcs PS, equal to AE; then

\[ 2AE : AOPD = AE : QP, \]

because AE is one-half of 2 AE, and QP one-half of AOPD. Consequently, since AOPD is a constant quantity, the force which makes the water oscillate is always proportional to the space which it runs through, and its oscillations are therefore isochronous. The force which makes the pendulum describe the arc PS, is to the weight of the pendulum, as PS is to PQ, or as AE is to PQ, since AE = PS; but the force which makes the water oscillate, is to the weight of the whole water in the same ratio; consequently, since the pendulum P, and the column AOPD, are influenced by the very same force, their oscillations must be performed in the same time. Q. E. D.

452. **Cor.**—As the oscillations of water and of pendulums are regulated by the same laws, if the oscillating column of water is increased or diminished, the time in which the oscillations are performed will increase or diminish in the subduplicate ratio of the length of the pendulum.

453. **Prop. II.**—The undulations of waves are performed in the same time as the oscillations of a pendulum whose length is equal to the breadth of a wave, or to the distance between two neighbouring cavities or eminences.

In the waves ABCDEF (fig. 76), the undulations are performed in such a manner, that the highest parts ACE become the lowest; and as the force which depresses the eminences A, C, E is always the weight of water contained in these eminences, it is obvious, that the undulations of waves are of the same kind as the undulations or oscillations of water in a siphon. It follows, therefore, from Prop. I., that if we take a pendulum, whose length is one-half BM, or half the distance between the highest and lowest parts of the wave, the highest parts of each wave will descend to the lowest parts during one oscillation of the pendulum, and in the time of another oscillation will again become the highest parts. The pendulum, therefore, will perform two oscillations in the time that each wave performs one undulation, that is, in the time that each wave describes the space AC or BD, between two neighbouring eminences or PART III.—ON HYDRAULIC MACHINERY.

454. To describe the various machines in which water is the impelling power, would be an endless and unprofitable task. Those machines which can be driven by wind, steam, and the force of men or horses, as well as they can be driven by water, do not properly belong to the science of hydraulics. By hydraulic machinery, therefore, we are to understand those various contrivances by which water can be employed as the impelling power of machinery; and those machines which are employed to raise water, or which could not operate without the assistance of that fluid.

455. A Hydraulic Machine will thus be one composed of a number of pieces, levers, or wheel-work, made up in such a way, that the movement impressed by water-power transmits itself from one part to another, and produces a certain result.

Hydraulic machines are of two kinds,—1st, Machines having a motion of rotation; and, 2d, Machines having an alternate motion. Under the first are comprehended hydraulic wheels, with turbines and reacting machines; and the second comprises water-column machines, and the hydraulic ram.

456. When a mover, as water, acts on a machine, it exercises on the part impinged on a pressure or effort: the immediate effect of this effort will be to make the part struck move in the direction of the power, or in some constrained direction; in either case space will be passed through by the part. Let \(a\) = the effect in lbs., and \(s\) = the space passed through, then \(a \cdot s\) = the sum of all the partial efforts developed by the machine, or the quantity of action developed by the mover in the operation; by some writers it is termed the mechanical work, or the work done.

Again, the dynamical force, or the force of a mover, will be the quantity of action developed in a unit or second of time; it will thus be the product of the effort and velocity, or, \(a \cdot s = a \cdot v\), since in 1 second \(s = v\).

457. Before, however, the force impressed on the palettes of a wheel can transmit its effect to the machinery to be set in motion, it experiences resistance from the pivots and from the air, friction from the teeth of wheels, and shocks of the teeth against the spindles. These resistances combined, make up a force which necessarily diminishes that originally derived from the mover. Hence, it is this diminished force that turns the machinery of a mill, and surmounts the resistance that the corn, for example, opposes to the grinding. The force necessary to overcome this resistance is named the useful effect; Montgolfier's definition of the same thing is—the active force is that which pays itself. There are, again, as it were inherent resistances connected with various pieces of the apparatus, which are called the passive resistances of the machine, the useful effect being taken for the active resistance surmounted. The total action of the passive resistances, together with the useful effect produced, make up the dynamical effect. It is clear, also, that the mechanical axiom, what is gained in power is lost in time or velocity, will hold good in hydraulics. Lastly, the total effect produced is equal to the force impressed on the machine.

CHAPTER I.—MACHINES HAVING A MOTION OF ROTATION.

Water-wheels have either a vertical or horizontal axis, and named respectively Horizontal and Vertical Wheels. The following table is a classification of machines having a motion of rotation:

| Vertical | Horizontal | |----------|------------| | with palettes | with palettes | | plane | curved | | in a course | circular | | rectilineal. Undershot Wheels. | Suspended Wheels. | | indefinite fluid. | Pneumatic Wheels. | | with buckets receiving water | at the summit. Overshot Wheels. | | under the summit. Breast Wheels. | placed in a cylinder. Tub or Spoon Mills. | | struck by an isolated vein. | out of a cylinder. Turbines of Fourneyron. | | with passages. Bardin's Turbine. | with reaction. Segner's Wheel or Barker's Mill; Euler, &c. |

SECTION I.—VERTICAL WHEELS, AXIS HORIZONTAL.

(a.) Undershot Wheels with plain Palettes.

458. Wheels with plain palettes are very simple in their parts and also in their construction. They generally consist of a horizontal axis, an inner and outer circle, on the latter of which perpendicular palettes or flat boards are fixed, and radii unite the several parts to the horizontal axis. Such a wheel (fig. 77) is set in motion by water conveyed to it by a course, which may be rectilineal or circular. In the former case, there is a flood-gate close to the wheel, collecting a head of water which is distributed by the gate when a very little uplifted, so that the water may just strike the lower palettes, and not pass between their edge and the course. In the latter we may have a course curved like that in fig. 77. The fall in that figure answers the same purpose as the flood-gate a little uplifted.

When a flood-gate is used, it should be set as near the wheel as possible, in order that the water might lose none of its velocity from the resistance of the bottom or sidewalls of the course. The opening of the flood-gate should be so disposed that the contraction of the water might be reduced to its least value; and the flood-gate itself should be inclined, which is the same thing as if the opening were brought nearer the wheel. It has been found that in a flood-gate inclined at an angle of 63° to the horizon, having 1 for base and 2 for height, the coefficient of contraction is 0·75; under an inclination of 45° the coefficient is 0·80, having 1 for base and 1 for height; and when the gate is vertical, in the same circumstances, it is about 0·70.

459. The course should be slightly inclined towards the wheel; its breadth should be determined by the volume of water which it is to convey; and the thickness of the fluid sheet escaping from under the flood-gate should never exceed 0·820 ft., nor be less than 0·492 ft. When it is smaller than this, so much water passes between the lower edge of the palettes and course, without producing any effect, thus diminishing the work of the water. The space between the walls of the course and the palettes should be from 0·032 ft. to 0·065 ft. The course should never be purely rectilineal; it should be somewhat inclined before reaching the flood-gate, where the water ought to strike the second palette at least from the vertical; then it must be concentrical with the exterior border of the palettes, and immediately after a palette has become vertical, the course should suddenly fall 3·93 inches at least; the water may then be conveyed away according to the locality. breadth of the course before reaching the palettes is a little narrower than the breadth of the wheel, but now it widens, and just embraces the wheel.

460. The breadth of the palettes will depend on that of the course, and on the magnitude of the intervals; their height will be about three times the thickness of the sheet issuing from under the flood-gate, never exceeding 2 ft. The distance between one board and another, measured on the periphery of the wheel, is a little less than their height. The number of palette-boards will depend on the diameter or circumference of the wheel. The dynamical effect of the wheel is not dependent on the diameter; it takes account of the velocity of the boards, or rather on the number of turns that the wheel makes in a certain time, so that the useful effect may be communicated to the machinery set in motion. Let \( u \) be the velocity per second of the extremity of the boards, \( n \) the number of turns in one minute; then 1 turn will be effected in the \( n \)th part of 1 min., or the diameter will be expressed by \( \frac{60}{n} = 19 \times \frac{n}{H} \). It has been found that, for a good effect, \( u = 17 \times \sqrt{H} \); and so the diameter of the wheel will be \( \frac{32}{n} \times \sqrt{H} \), where \( H \) is the total fall of water. Generally, the diameter is never less than 13 ft., and never in excess of 26 ft.

With respect to the number of palettes, artificers generally adopt the rule, that—

| Under a diameter of 13 ft. | the number of boards is 28 | |---------------------------|--------------------------| | ... | ... | | ... | ... |

and so on. Four boards being added for every 3 feet of breadth to the diameter of the wheel.

461. It is not advantageous to have the palettes inclined to the direction of the radius, when the water is conveyed by a course, for Bossut found that when he inclined the boards at angles of 0°, 8°, 12°, and 16°, the effects were respectively as the numbers 1, 0.949, 0.956, 0.998. Inclined paddles are very useful when plunged in an indefinite fluid.

462. In order that such a wheel may be put in motion, the water leaves the opening under the flood-gate with a velocity due to the head pent up behind the flood-gate; this water impinging on the palettes with a considerable pressure, imparts a movement of rotation to the wheel. But this velocity depends on the resistance to be overcome, so that if the resistance be considerable, the velocity will be lessened. Again the pressure exercised by the water on the boards, is not the same when they are moving as when they are at rest, for the pressure is feebler in proportion as the wheel moves quicker. Hence, therefore, the wheel must take a particular velocity by overcoming a given resistance, that the pressure on the boards may correspond with this resistance. The wheel, however, may be made to take any velocity, by regulating the resistance which it has to overcome; but the amount of work given to the wheel by the water will not always be the same unless the wheel turn with a certain velocity. This velocity must be neither too great nor too small; it must be such that the work done by the wheel will exceed that which it would produce with any other velocity. Experiment has shown that this maximum work is attained when the velocity of the wheel, measured by its circumference, is 0.45 times that of the water on arriving at the palette boards. When the wheel has this velocity, it is found that the work done by it does not exceed 0.25 the work of the water, so that nearly 0.75 of this latter work is lost.

463. It is very important, for those especially who employ water-wheel power, to know the work which their wheels are capable of giving out. The general equation for this kind of wheel is the following:—Let a constant stream impinge perpendicularly on the boards of a water-wheel moving with a certain velocity, \( W \) being the weight of a current thrown on the paddle-board every second; \( V \) and \( V' \) the velocity of the water on striking and on leaving the board; and \( V'' \) that of the moving board; then the velocity lost will be \( V - V'' \), and the work lost by impact

\[ P = \frac{W}{g} (V - V'')^2 \]

and the work which the stream is capable of giving out

\[ P = \frac{W}{g} V^2 - \frac{W}{g} (V - V')^2 = \frac{W}{g} \left[ V^2 - (V - V')^2 \right] = \]

\[ P + \frac{W}{g} V'^2 \]

which last is the useful work + the work remaining in the water after leaving the paddle.

464. In the case of this Undershoot wheel, \( P = \frac{W}{g} (V - V')V \), for \( V' \) only is variable and \( = V' \); if \( V' = 0 \), the wheel does no useful work, and so also when \( V = V' \); also \( V' \) can never exceed \( V \). Thus the effect increases as the velocity of the wheel, starting from zero, increases, but only up to a certain point, for it evidently diminishes when it approaches \( V \). Hence, there is a maximum effect between these two extremes, that is to say, \( (V - V')V' \) is to be a maximum;

\[ V \cdot dV' - 2V' \cdot dV = 0; \quad \text{or} \quad V' = \frac{1}{2} \cdot V; \]

or an undershot wheel produces its utmost effect when its velocity is one-half that of the current.

Borda first showed the truth of the above result; it has been fully confirmed by the experiments of Smeaton, Bosset, and others.

465. Many experiments have been made on undershot wheels. Smeaton has given the results which he obtained in the Royal Society Trans. for 1759. It is supposed that his model wheel was of too small dimensions to give very accurate results for practice. The value which Smeaton allowed for the velocity of the wheel, or \( V \), was \( V = 0.40 \times V \). Bossut reckoned it of the same value. More recent experiments, however, regard it as \( 0.45 \times V \) for the case of the maximum effect. The coefficient also for reducing the theoretical to the real effect may be taken as \( 0.6 \). Smeaton's was 0.64. It is necessary, therefore, to multiply the expression \( P = \frac{W}{g} (V - V')V' \) by \( m \), the coefficient, to obtain the true result.

466. When an undershot wheel has a fall, the float-boards disposed on its circumference receive the impulse of the water conveyed to the lowest point of the wheel by an inclined canal. It is represented in fig. 77, where WW is the water-wheel, and ABDFHKMV the canal or mill-course, which conveys the water to K, where it strikes the plane float-boards &c., &c., and makes the wheel revolve about its axis.

467. In order to construct the mill-course to the greatest advantage, we must give but a very small declivity to the canal which conducts the water from the river. It will On Water Wheels.

be sufficient to make A B slope about one inch in 200 yards, making the declivity, however, about half an inch for the first 48 yards, in order that the water may have sufficient velocity to prevent it from falling back into the river. The inclination of the fall, represented by the angle GCR, should be 25° 50', or CR, the radius, should be to GR, the tangent of this angle, as 100 to 48, or as 25 to 12; and since the surface of the water Sg is bent from ab into ac before it is precipitated down the fall, it will be necessary to incarve the upper part BCD of the course into BD, that the water in the bottom may move parallel to the water at the surface of the stream. For this purpose, take the points B, D, about 12 inches distance from C, and raise the perpendiculars BE, DE. The point of intersection E will be the centre from which the arc BD is to be described; the radius being about 10 1/2 inches. Now, in order that the water may act more advantageously upon the float-boards of the wheel WW, it must assume a horizontal direction, with the same velocity which it would have acquired when it came to the point G. But, if the water were allowed to fall from C to G, it would dash upon the horizontal part HG, and thus lose a great part of its velocity. It will be necessary, therefore, to make it move along FH, an arc of a circle to which DF and GH are tangents in the points F and H. For this purpose make GF and GH each equal to three feet; and raise the perpendiculars HI, FI, which will intersect one another in the point I, distant about 4 feet 9 inches from the points F and H, and the centre of the arc FH will be determined. The distance HK, through which the water runs before it acts upon the wheel, should not be less than 2 or 3 feet, in order that the different filaments of the fluid may have attained a horizontal direction. If HK were too large, the stream would suffer a diminution of velocity by its friction on the bottom of the course. That no water may escape between the bottom of the course KH and the extremities of the float-boards, KL should be about 3 inches, and the extremity o of the float-board so ought to reach below the line HKX, sufficient room being left between o and M for the play of the wheel; or KLM may be formed into the arc of a circle KM, concentric with the wheel. The line LMV, which has been called the course of impulsion, should be prolonged so as to support the water as long as it can act upon the float-boards, and should be about 9 inches distant from OP, a horizontal line passing through O, the lowest point of the fall; for, if OL were much less than 9 inches, the water, having spent the greatest part of its force in impelling the float-board, would accumulate below the wheel, and retard its motion. For the same reason another course, which has been called the course of discharge, should be connected with LMV by the curve VN, to preserve the remaining velocity of the water, which would otherwise be discharged by falling perpendicularly from V to N. The course of discharge which is represented by the line VZ, sloping from the point O, should be about 16 yards long, having an inch of declivity for every 2 yards. The canal which reconducts the water from the course of discharge to the river should slope about 4 inches in the first 200 yards, 3 inches in the second 200 yards, decreasing gradually till it terminates in the river. But if the river to which the water is conveyed should, when swollen by the rains, force the water back upon the wheel, the canal must have a greater declivity to prevent this from taking place. Hence it is evident that very accurate levelling is requisite to the proper formation of the mill-course.

(β.) Breast or Side Wheels, with plain Palettes.

468. If we suppose that the wheel of fig. 77 is moved so far to the right, that the extreme edges of its boards, when turning round, will all but graze the circular part of the course, and if the walls of the passage box in the wheel so that no water will escape between the palettes and the passage, or at the side walls, we shall have a Breast or Side wheel. These breast-wheels are divided into High Breast or Low Breast, according as they receive the water above or below their horizontal diameter. The breast-wheel, fig. 78, is driven partly by the impulse, but chiefly by the weight of the water. The water may be received on the palettes, as in fig. 77, or by means of a flood-

gate, or by means of a vertical passage communicating with a reservoir at a height above the wheel; in either case the water will be delivered on the plane-boards of the wheel.

In fig. 78 the water passes through an iron grating ab, and its admission is regulated by two shutters c, d, the lowest of which is adjusted till a sufficient volume of water passes over it; and the other shutter, e, is made to descend by machinery when the wheel is to be stopped, and the water retained in the reservoir R. When the water enters on any board, it is generally retained there till it arrives at the lowest part of the wheel, and is discharged into the course.

469. Mr Lambert, of the Academy of Sciences of Berlin, experimented on these wheels, and observes, that a breast-wheel should be used when the fall of water is above four feet high and below ten. M. Morin has recently made experiments on this wheel, which may be seen in his Expériences sur les Roues Hydroliques, &c. It appears from experiment that the diameter of these breast-wheels should vary from 16 to 23 ft. The number of palettes is found from art. 460, and generally they are a little inclined to the radius.

470. In determining the formula for wheels having a fall of water, we have to take into account the additional height that the water descends with the wheel, where, consequently, an additional amount of work will be performed. Let this amount of work be that due to the vertical space h, through which the water falls. Then, work applied to the wheel in one second = \[ \frac{W}{2g} \left\{ V^2 - (V - V')^2 \right\} + Wh \]

= the useful work P performed by the wheel + that due to the water on leaving the wheel. Let V' be the velocity of the water on leaving the board; its effective work will be \[ \frac{W \cdot V'^2}{2g} \]

∴ \( P + \frac{W \cdot V'^2}{2g} = \frac{W}{2g} \left\{ V^2 - (V - V')^2 \right\} + W \cdot h; \)

or, \( P = W \cdot h + \frac{W}{2g} \left\{ 2VV' - V'^2 - V^2 \right\}; \)

---

1 Nouv. Mémo. de l'Académie de Berlin, 1775. which is the general equation for the effective work of a water-wheel when the water has a fall.

In the case, therefore, of the Breast-wheel, \( h \) is the vertical height of the fall where the water meets the paddle, and \( V' = V \);

\[ P = W \left( h + \frac{1}{g} (V - V') V' \right). \]

The latter member is to be multiplied by the coefficient, which, according to M. Morin, is 0·74.

This wheel may be wrought very efficiently with velocities varying from \( \frac{1}{2} V \) to \( \frac{3}{2} V \).

471. Suspended wheels are such as work in an indefinite course of water. The plane palettes are plunged in the current of a river, the force of which acting on the immersed boards, communicates a rotatory motion to the wheel. They are generally to be seen established on the sides of boats, or other supports properly fixed. The diameter of these wheels never exceeds 16 ft. The palette-boards may number 12, 18, or even with advantage 24; their height may be about the 4th part or more of the entire radius of the wheel, and the breadth from 8 to 16 ft.

From experiments on this kind of wheel, it appears that the velocity of the palettes, at the middle of their height, should be, for the greatest possible effect, 0·10 that of the water.

(γ.) Undershot Wheels with curved Palettes—Poncelet Wheel.

472. The undershot wheel already mentioned in art. 458 loses a considerable amount of the force impressed upon it by water, and so the useful effect is much reduced. It was of importance to have a wheel possessing the conveniences of the undershot wheel, but avoiding the loss of force which is always incident to the use of it. Such a wheel is the Poncelet wheel, devised by M. Poncelet in 1824. This wheel, represented in fig. 79, is an undershot wheel, but, unlike that in fig. 78, has curved palettes; the curves are tangents to the exterior rim of the wheel; the lower element of curvature is horizontal, the upper vertical. It is with boards so disposed that the work transmitted by the water to the wheel exceeds that given out by plain boards.

473. The course, till arriving at the lowest part of the wheel, is here inclined at about 1 in 10, so that the water might regain the velocity which it lost by the friction of the sides and bottom of the course: the inclination may be lessened, when the sheet of water is considerable that issues from under the flood-gate. The course also is uniform, and where it embraces the wheel is a little narrower than its breadth. In order that the wheel may work in a course narrower than itself, the lower parts of the upright sides of the course are hollowed out, so that the projecting curved boards of the wheel might work in them. At the lowest part of the wheel the course is suddenly depressed, in order that the back-water might not affect the working of the machine. With respect to the head of water pent up behind the flood-gate, the latter is very much inclined, so that the sheet of escaping water might just issue without losing any of its velocity by the resistance of the course.

474. If we suppose that water is admitted by the opening on the curved boards, it will take a velocity, say \( V \), and enter horizontally on the lower element of the curve. The water now glides along this curve, rises, and gradually, by the action of gravity, loses its velocity of entering, till, at the highest part of its course, \( V \) is lost altogether, that is, when it has attained a height \( \frac{V^2}{2g} = 0·051 \text{ met.} \times V^2 \). The water now begins to fall down the curve, and, acquiring an increased velocity as it reaches the horizontal or lowest part of the curve, quits the palette with the same velocity as it entered upon it. Let the uniform velocity of the perimeter of the wheel be \( V \); then \( V - V' \) is the relative velocity with which the water rises through the length of the curve, and \( \frac{(V - V')^2}{2g} = 0·051 \text{ met.} \times (V - V')^2 \) is the height to which it will rise on the curve; and at last, on quitting the horizontal part of the board, will have a velocity \( V - V' \). But the lower part of the curve works with a velocity \( V' \) in the opposite direction; \( V - V' - V' = V - 2V' \) is the absolute velocity of the water as it leaves the wheel. If we wish to have a maximum effect, \( V - 2V' = 0 \), or \( V = 2V' \); that is to say, were the wheel to take one-half the velocity of the current on its arrival at the board, the absolute velocity of the water on quitting the boards would be nothing.

Wherefore we have a current mover, experiencing neither shock nor loss of velocity at the moment of its entering on the wheel, and which possesses none at the moment of leaving it; the current then spends its whole motion, and communicates its whole force to the wheel. These two conditions suffice to produce the greatest possible effect. For this let \( W \) be the weight of fluid furnished by the current in 1 second, and \( h \) = height due to velocity, the effect will be expressed by \( Wh \). This is true for a single fillet, but not for a sheet of water of a definite thickness; for the molecules strike the boards at an angle more or less great with respect to the parts struck, and consequently there is a loss of force and velocity. This mass of water, when it quits the boards, does not move exactly in a direction opposite to the boards; and again, since so much water is lost by every course, without producing any effect, the real effect will be something less than \( Wh \).

475. M. Poncelet, in his Memoire sur les Roues Hydrauliques Verticales a Aules Courbes, mentions all the experiments which he made on this wheel. The practical results which Poncelet deduced from these experiments are—1st. That the velocity of the wheel which gives a maximum effect is 0·55 of the velocity of the current; it may vary from 0·50 to 0·60. 2d. That the dynamical effect is not under 0·75 × \( Wh \) for small falls with large openings at the bottom of the flood-gate, and 0·65 for small openings and deep falls. 3d. That this same effect, compared with the full force of the mover, is 0·60; and for small openings it will reach 0·50.

476. For the case which generally occurs in practice, when properly arranged wheels have a velocity not differing much from 0·55 that of the current, we may take the dynamical effect = 0·75 × \( Wh \), and also = 0·60 × \( WH \), the passive resistances being taken into account. \( H \) is the total fall of water.

With an opening under the flood-gate of from 0·100

---

1 Essai sur la Construction des Roues Hydrauliques, etc., par M. Fabre. On Water, met. to O-304 met., h will vary from 1-43 to 1-26 met., and H Wheels, from 1-59 met. to 1-52 met.; and the weight of water W, furnished by the current in 1 second, would be, for the first opening 279 kilogr., and for the second 809 kilogr.

The coefficient which multiplies the quantity Wh, is fully double that obtained from wheels with plain palettes, which is sufficient to make us dispense with plain palette-wheels, and substitute for them wheels with curved palettes.

477. In the construction of these wheels with curved palettes—1st. The number of palettes will be double that required in wheels with plain palettes. 2d. Their height, in the direction of the radius, or the distance between the inner and outer circumference of the wheel, must be in excess of a fourth of the effective fall; if the fall be 1-40 met., it will be a third of this; for falls under this last it will be a half. 3d. The lower element of the curve, which makes an angle nothing, or nearly so, with the outer circumference, will, when the sheet of water is extremely thin, make an angle with it of 24°, 30°, and generally greater as the thickness of the moving sheet increases. The curves may be thus obtained.—From point A (fig. 79), where the fluid sheet BA meets the outer circumference, raise a vertical AK, and from point C, where it cuts the inner circumference, describe, from centre C, with radius CA, an arc AE; and so on for all the others. And 4th, Immediately under the vertical diameter, the bottom of the course must be suddenly lowered, so that the back-water might not interrupt the working of the wheel.

478. The work done then by the water entering on the Poncelet wheel = work done on the wheel + that done by the water on leaving the paddle, or

\[ \frac{W}{2g} V^2 = P + \frac{W}{2g} (V - 2V)^2. \]

\[ \therefore P = \frac{2W}{g} (V - V') V'; \]

and we have seen that the work done by this wheel is nearly double that performed by the undershot wheel with plain palettes; the value of P being as in art. 470.

(δ.) Overshot Wheels, or Bucket Wheels.

479. An overshot or bucket wheel is so disposed, that the water, as the motive power, is received on its upper part; it is represented in fig. 80, where ABC is the circumference of the wheel furnished with a number of buckets. The canal MN conveys the water into the second bucket from the top Aa. The equilibrium of the wheel is therefore destroyed; and the power of the bucket Aa to turn the wheel round its centre of motion O, is the same as if the weight of the water in the bucket were suspended at m, the extremity of the lever Om; c being the centre of gravity of the bucket, and Om a perpendicular let fall from the fulcrum O to the direction cm, in which the force is exerted. In consequence of this destruction of equilibrium, the wheel will move round in the direction AB, the bucket Aa will be at d, and the empty bucket b will take the place of Aa, and receive water from the spout N. The force acting on the wheel is now the water in the bucket d, acting with a lever mO, and the water in the bucket Aa acting with a lever mO. The velocity of the wheel will therefore increase with the number of loaded buckets, and with their distance from the vertex of the wheel; for the lever by which they tend to turn the wheel about its axis increases as the buckets approach to e, where their power, represented by eO, is a maximum. After the buckets have passed e, the lever by which they act gradually diminishes, they lose by degrees a small portion of their water; and as soon as they reach P, it is completely discharged. When the wheel begins to move, its velocity will increase rapidly till the quadrant of buckets be is completely filled. While these buckets are descending through the inferior quadrant eP, and the buckets on the left hand of b are receiving water from the spout, the velocity of the wheel will still increase; but the increments of velocity will be smaller and smaller, since the levers by which the inferior buckets act are gradually diminishing. As soon as the highest bucket Ac has reached the point B, where it is emptied, nearly the whole semi-circumference of the wheel is loaded with water; and when the bucket at B is discharging its contents, the bucket at A is filling, so that the load in the buckets, by which the wheel is impelled, will be always the same, and the velocity of the wheel will become uniform.

480. In order to determine the best form of the buckets, we must consider that the power of the wheel would be a maximum if the whole of its semi-circumference were loaded with water. This effect would be obtained if the buckets had the shape shown in fig. 81, where ABC is the form of the bucket, AB being a continuation of the radius, and BC part of the circumference of the wheel, and nearly equal to AD. But as a small aperture at CE will neither admit nor discharge the water, the form shown in fig. 82 has been proposed by Sir David Brewster as the best. In this construction, BC is made a little larger than BE, and AB is diminished so as to make the angle ABC a little greater than 90°. The angles at B should be rounded off, so as to make ABC a curve, as indicated by the dotted line. The aperture at dE must be sufficient for the introduction and discharge of the water, and the side BE of the bucket should be as smooth and even as possible.

481. The distance of one bucket from another is about one foot. But the wheel is sometimes divided into four parts, and a whole number of buckets placed in each, so that the number of buckets will be proportional to the diameter. In forming the buckets, the space dE, fig. 82, must evidently be a little wider than the fluid sheet, so that the proper volume of water may be admitted. The angle also which the side AB makes with BE, is from 110° to 118°, according as the wheels are from about 13 to 39 feet in diameter; the angle then which BE makes with the circumference, is about 31°; it should never exceed 33°. The depth of the bucket is DE, where E is a point On Water of the circumference, fig. 81. AB is frequently taken as a third of DE, sometimes as a half. The buckets may receive water either by the method shown in fig. 80; or the water, without fall of any kind, may come in the opposite direction by a canal closed up at the end next the wheel, where one or two openings in the bottom of the course admit the water into the buckets. This principle is in very general use in Great Britain.

482. The breadth of the bucket is determined by the volume of water which it receives and carries. Let Q be the volume issuing in one second from the reservoir by which MN is supplied. V' the velocity of the circumference, d the distance, reckoned along AB (fig. 80), of one bucket from another; then in one second a number of buckets \( \frac{V'}{d} \) will pass before the mouth of MN, and each will carry away a volume \( \frac{Q}{d} \) divided by \( \frac{V'}{d} = \frac{Qd}{V'} \). Now, in order to be effective, the bucket must not only carry this, but a volume three times greater. Let l = the breadth of a bucket, s = the transverse section, then the capacity of the bucket or volume of water contained \( = sl \). Now, let m be the number of buckets on the wheel, and n the revolutions of the wheel in one minute, or 60 seconds; then as the volume \( sl \) of water must be \( = 3Q \frac{d}{V'} = 180 \frac{Q}{m.n} \); for \( m.d = \) circumference, and 60 \( V' = n \times \) circumference;

\[ sl = 180 \frac{Q}{m.n} ; \quad \text{or} \quad l = 180 \frac{Q}{m.n.s} \]

breadth of wheel given by the work. The volume Q is that which the wheel spends so as to produce its whole effect.

483. In the work performed by an overshot wheel, the mean vertical height h is nearly equal to the diameter of the wheel, and also \( V' = V \), for the velocity of the water on leaving the paddle is the same as that of the paddle.

\[ P = Wh + \frac{W}{2g} \left( 2VV' - V'^2 - V^2 \right) \]

\[ = W \left( h + \frac{1}{g}(V - V')V' \right). \]

The latter member is to be multiplied by its coefficient.

484. Smeaton, in 1759, experimented on overshot wheels, and the principal results of his observations are—1st. That, relative to the effect produced, the fall should be divided into two parts; one equal to the diameter, the other a little above it. 2d. Since the action of the upper part of the fall is feeble, Smeaton sought a relation between the effect and the lower part of the fall; this was constantly 0°80 × W × diameter. 3d. When the fall exceeded the diameter by only a small quantity, he had 0°72 × W × total fall. And, 4th. That the velocity of a bucket-wheel should range from 3 feet to 6 feet per sec. In these observations he was guided mainly by the old principle, that a bucket-wheel gives most effect when it turns slowly, as was shown in the experiments of Deparcieux. Smeaton also concluded, from a series of experiments, that the higher the wheel is in proportion to the whole descent, the greater will be the effect.

Among other philosophers who have experimented on overshot wheels, we may mention the Chevalier D'Arcy, who showed that the less the velocity the greater is the effect produced; Borda, who showed that the maximum effect will be produced when the diameter equals the height of fall; and Bossut, Albert Euler, Lambert, Poncelet. Smeaton also showed, that when the work performed was a maximum, the ratio of the power to the effect was as four to three, when the height of the fall and the quantities of water expended were the least; but that it was as four to two, when the height of the fall and the quantities discharged were the greatest. By taking a mean between these ratios, we may conclude, in general, that in overshot wheels the power is to the effect as three to two. In this case the power is supposed to be computed from the whole height of the fall; because the water must be raised to that height in order to be in a condition of producing the same effect a second time. When the power of the water is estimated only from the height of the wheel, the ratio of the power to the effect was more constant, being nearly as five to four. According to Smeaton, the effect of a wheel driven in this manner is equal to the effect of an undershot wheel (458), whose head of water is equal to the difference of level between the surface of water in the reservoir, and the point where it strikes the wheel, added to that of an overshot (479), whose height is equal to the difference of level between the point where it strikes the wheel and the level of the tail water.

485. We have hitherto supposed the float-boards, though inclined to the radius, to be perpendicular to the plane of the wheel. Undershot wheels, however, have sometimes been constructed with float-boards inclined to the plane of the wheel. A wheel of this kind is represented in fig. 83, where AB is the wheel, and C, D, E, F, G, H, the oblique float-boards. The horizontal current MN is delivered on the float-boards, so as to strike them perpendicularly. On account of the size of the float-boards, every filament of the water contributes to turn the wheel; and therefore its effect will be greater than in undershot wheels of the common form. Albert Euler imagines that the effect will be twice as great, and observes, that in order to produce such an effect, the velocity of the centre of impression should be to the velocity of the water, as radius is to triple the sine of the angle by which the float-boards are inclined to the plane of the wheel. If this inclination, therefore, be 60°, the velocity of the wheel at the centre of impression ought to be to the velocity of the impelling fluid as 1 to \( \frac{3\sqrt{3}}{2} \), that is, as 5 to 13 nearly, because \( \sin 60° = \frac{\sqrt{3}}{2} \). When the inclination is 30°, the ratio of the velocities will be found to be as 2 to 3.

SECTION II.—HORIZONTAL WHEELS, AXIS VERTICAL.

(a) Wheels receiving an isolated Vein.

486. Horizontal water-wheels have been much used on the Continent, and are strongly recommended to our notice by the simplicity of their construction. In fig. 84, AB is the large water-wheel which moves horizontally upon its... Horizontal arbor CD. This arbor passes through the immovable mill-stone EF at D, and being fixed to the upper one GH, carries it once round for every revolution of the great wheel.

The mill-course is constructed in the same manner for horizontal as for vertical wheels; it must be circular and concentric with the rim of the wheel, sufficient room being left between it and the tips of the float-boards for the play of the wheel. In this construction, where the water moves in a horizontal direction before it strikes the wheel, the float-boards should be inclined about 25° to the plane of the wheel, and the same number of degrees to the radius, so that the lowest and outermost sides of the float-boards may be farthest up the stream.

(B.) Tub or Spoon Wheels.

487. In many parts of the south of France, wheels are to be seen working in a cylindrical tub or hole of small depth, or into a conical hollow. Fig. 85 will show the nature of one of these wheels. It is constructed in the form of an inverted cone AB, with spiral palettes on its surface. The wheel moves on a vertical axis AB, and is driven chiefly by the impulse of water conveyed by the canal C, and falling obliquely on the float-boards at the place of impact: the spent water is carried away by the channel M in the solid walls D, D. But in the best constructed cylindrical tub-wheels the water issues from a canal in the walls, in a direction tangential to the circumference. In thus impinging on the float-boards, it preserves its gyratory motion, and makes the wheel whirl round. The useful effect of these wheels is never above 0'25 of the work given by the water.

Spoon-wheels are so named because their narrow, yet long, radial curved palettes, have a shape like a spoon.

(C.) Turbine of M. Fourneyron.

488. Within the last 30 years wheels with vertical axes have been so much improved, that now they rank among the most perfect hydraulic machines which exist. They are termed Turbines, and were invented in 1827 by a young French mechanician named Fourneyron. During that year, Fourneyron erected one a few inches in diameter, of six horse-power, in Franche-Comté. To the surprise of all, when the machine was put in action, the effective work produced was 0'80 that of the water. The surprise was not diminished when it was found that the new wheel worked as well under as above water. Since their invention, turbines have been extensively adopted, especially in France and Germany.

489. A turbine consists of three principal parts; the turbine proper DD (fig. 86), with its axis FF; the cylinder CC, with its bottom CC and the apparatus aa which carries it; and the cylindrical flood-gate aa. The whole apparatus is made of metal.

Turbines are of two kinds—those which work below, and those which work above, the surface of the water.

490. With respect to the first kind, or those immersed in water, in fig. 86 we have AA the upper or supplying canal, the water from which enters freely the cylinder BB, and finding no way of escape save by the circular lateral opening CC, issues by it and precipitates itself in every direction in a sheet upon the palette boards of the circular wheel DD. It thus puts the wheel in motion, and then loses itself in the lower water-course. The wheel DD is annular, and disposed horizontally all round the opening CC, such that the fluid sheet might impinge upon it. We shall have a proper idea of the disposition of the wheel DD if the Poncelet wheel, instead of working vertically, work horizontally; we have a section of it, however, in fig. 87.

A kind of metal cap EE, binds the wheel to the central revolving axis FF, passing freely through a pipe in the middle of the cylinder. The wheel DD is in the present case wholly immersed in the lower water-course, the level of which is GG. The axial beam FF of the machine is terminated at its lower part by a pivot resting on a point of a lever HK, of which the fulcrum is K. A rod L is jointed to H the extremity of the lever, and the end of that rod works into a nut; on turning the nut, we may raise or lower at pleasure the axial beam, and therefore also the wheel which it carries, so that the latter may be brought exactly in opposition to the circular opening of the cylinder in order to receive the due supply of water. Although the wheel is immersed in the lower water-course, yet this does not prevent the water of the higher course, i.e., of the cylinder, from passing through the opening CC, and acting on the boards of the wheel, the flow being due to the difference of the levels of the two water-courses. Again, there must be some arrangement within the cylindrical reservoir, so that the water which flows in may flow out in a uniform equable sheet; if this were not attended to, the escape would take place at different points, and would therefore be irregular; and, further, the water would not strike the boards of the wheel in the most effective manner. In order, therefore, to obviate this irregularity of the flow, M. Fourneyron disposed within the cylinder curved upright partitions B, as in fig. 87, which is the projection of the interior of the cylinder on its bottom; at a small distance from this, and disconnected with it, is the working wheel D, with its curved boards disposed opposite to the curved partitions of B. The water enters and fills the chambers of B, and the consequence is, that the water, on issuing from the reservoir, moves everywhere obliquely to the surface of the same, encounters the boards of the wheel which offer an opposition to its motion, and exercises on the palettes, pressures which tend to turn the wheel in the direction of the arrow. Within the reservoir is a cylindrical flood-gate aa (fig. 86), which serves to widen or narrow the circular opening CC. This is done by raising or lowering the vertical rods b, b, their upper parts working into nuts which may be turned at pleasure. The flood-gate aa is of considerable thickness, and round, so as to widen the orifice of escape.

It would appear that the curved boards of the wheels presenting their surfaces nearly perpendicularly to the issuing water, ought to receive a shock from part of the liquid, but this entirely disappears when the wheel is moving in a proper manner. It must be observed, however, that these boards moving round a centre, are differently circumstanced than if they had been at rest. Whenever the water issues out of the reservoir it strikes on the boards, but no sooner does this take place than the wheel flies before the water; the palette boards are only acted upon in virtue of the relative velocity which the water possesses with respect to the boards. Now the boards are so disposed, that when the turbine assumes the velocity that it ought to take, the relative velocity of the water with respect to the wheel is directed tangentially to its inner edge; the result is that the water enters on the wheel without producing any shock. The water moves along the whole length of the wheel boards, entering at the inner and passing out by the exterior part of the board, and exercising a pressure at every point, whereby the velocity is constantly changing its direction. The water also, on leaving the wheel, leaves it with a relative velocity, and moves in a direction contrary to that of the boards. It is possible, then, that the turbine may be made to have a motion such that the velocity of the exterior circumference may be precisely the same as the relative velocity of the water. If this condition be fulfilled, the water, on its leaving the wheel, will only be animated by an insensible motion, and will thus become associated with that in the midst of which the wheel is plunged; the water will thus be deposited, so to speak, without velocity from the boards, which fly away without being retarded.

We see also that the water, acting at the same time on all the boards of the wheel, produces on these, horizontal pressures which tend in no ways to throw the axis of the wheel to one side or another; the consequence of which will be that the pressures cause no friction of the beam on its pivot, nor on the bodies which touch it at different parts of its height, and which tend to keep it in an exact vertical position. These conditions, which no wheel with axis horizontal can fulfil, enables the turbine to give results far more valuable than those of Poncelet's wheel.

From the various experiments which have been made on horizontal turbines, the greatest amount of work done equals 0.75, and in some cases 0.85 of that expended by the water.

491. The Fourneyron turbine has several advantages of the most valuable kind. It may work when immersed, as we have seen, in the middle of the lower course, and thus disposed, it operates during the time of floods as well as when the water is very low, for otherwise it would be disturbed by the greater or lesser height of the level of the water in the lower course; the total height of fall is also rendered useful, which could not be the case were the wheel to be placed above the level of the lower water-course; and that the machine can work during hard frost, since the water seldom freezes in a rapid running stream. Another advantage which this machine has, consists in this, that we may make its velocity vary within sufficiently extensive limits, by means of the velocity which corresponds to the maximum effect, without which the ratio of work done to the work which the quantity of water expended represents, would diminish very much. This result is of great importance for the case in which a turbine ought to move with a uniform velocity, and where the height of the falling water varies. But the velocity of a turbine which corresponds to the maximum effect, depends on the height of the fall; it augments or diminishes according to this height. If a turbine move always with the same velocity under different heights of fall, it will not have a constant velocity capable of producing the maximum effect. It is then a very important matter that the machine, acting with a velocity different from the particular velocity, should furnish results nearly equal to the maximum effect.

The turbine may be adapted for all sorts of fall, provided that we adapt it to the quantity of water, more or less great, which ought to act upon it, and to the rapidity of motion which it ought to take. We have an idea of the value of a properly constructed turbine, when one erected in 1837 by M. Fourneyron, at St Blaise, in the Black Forest of Baden, moved by a column of water 172 feet, did work equivalent to 56 horse-power, and gave a useful effect of between 0.70 and 0.75 of that of the water; the wheel, together with the cylinder, which is 20 inches in diameter, weighed only 105 lbs. Another at the same place has a fall of 354 feet; the diameter of the wheel is 13 inches; it expends a volume nearly of 1 cubic foot per second, and makes from 2200 to 2300 revolutions per minute; its useful effect is from 80 to 85 per cent. of the work derived by the water. This machine sets in motion 8000 water spindles, with roving, carding engines, cleansers, &c. A turbine erected at Gisors, under a fall of 37 feet, utilized 0.75 the work of the fall. When the fall was 2 feet, the useful work was 66 per cent. of the fall, and when the fall was 111 feet, it was 60 per cent.

492. The second kind of turbines, or those which work out of the water, is represented in fig. 88, where B is the entire cylinder; C is that part of it which joins on to the supplying pipe D, conveying the water from an ele- vated reservoir or canal; AA is the turbine or wheel; a, is the circular flood-gate, which may be lowered or elevated by the rods b, b.

493. With all the advantages, however, which the Fourneyron turbine has, there is one thing which causes so much loss of work in the machine; the sheet of water issuing from the circular opening at the lower side of the reservoir, has a thickness more or less great, according to the contraction of the flood-gate; on this account the water does not always fill the wheel in its whole height. The upper part of the space comprised between the boards of the wheel is not, however, empty; but the water there does not possess the same velocity as that issuing from the reservoir; this occasions eddies, accompanied by a loss of velocity, which must therefore produce a diminution of useful effect. It was to remedy this waste that Fourneyron divided his wheel into several compartments, by means of horizontal partitions. But these partitions do not recover the whole, though a part, of the loss to the machine.

M. Callon endeavoured, by the following method, to restore the full effect of the machine. To this end he substituted for the single circular flood-gate within the reservoir of Fourneyron, a considerable number of partial flood-gates, all touching each other, and placed around the inside of the cylinder, several being open that the water might issue upon the wheel. By this means, the quantity of water flowing from the reservoir may be diminished without diminishing the thickness of the sheet; those of the flood-gates which are open and shut being uniformly distributed around the circumference. The disadvantage of Fourneyron's turbine is thus removed, but another is originated in the fact, that the different portions of the wheel are carried round successively before the open and before the shut flood-gates. At the time when the interval of two boards comes opposite a shut gate, the water which is there contained, and which has a considerable velocity, will only continue to move on producing a void behind it, which will necessarily occasion a sharp diminution in its velocity, and, therefore, also will entail a loss of work.

494. Fontaine's turbine differs from that of Fourneyron's in having the circular opening not in the side near the bottom, as in the latter, but in the bottom of the cylinder itself, with the wheel right under the same. The circular opening is not wholly open, but it consists of a considerable number of distinct orifices, each of which is furnished with a special flood-gate, by the aid of which the orifice may be more or less shut. This disposition of the machine does away in great part with the loss which takes place in Fourneyron's turbine.

495. The turbines which have been mentioned hitherto are difficult to repair, both when immersed, and when at a small distance above the lower water-course. The method of doing so is by raising a strong temporary barricade around the turbine, and pumping the water out so as to isolate it, the cylinder remaining full, while the flood-gate is firmly secured to prevent escape. But this method is inconvenient. A turbine, therefore, was invented by M. Jonval, and constructed and improved by M. Kechlin for the special purpose of remedying this inconvenience. Kechlin's turbine is somewhat peculiar. Let us suppose a hollow vertical cylinder open at both ends, and the upper and lower water-courses communicating with each other by means of this cylinder; the water which passes through this cylinder may be rendered useful by placing a turbine wheel within it, and at any part of its height; provided only that the water, on leaving the wheel, and traversing afterwards that portion of the cylinder lying between the bottom of the wheel and the surface of the lower water-course, is not in direct contact with the atmosphere till it arrives at the surface of the lower water-course. Now, it is clear that if the turbine be placed too high in the cylinder, the difference of level between the upper course and the wheel will be small, and no horizontal force will be lost; but to make up for the loss, a gain is effected by the drawing or sucking, which takes place in the lower part of the cylinder under the wheel; this drawing effect being, very curiously, greatest when the wheel is highest above the surface of the lower water-course. We see, then, that the position which the machine now has, allows a workman to enter the cylinder from below, and repair the wheel.

For an account of the best observations and experiments on the turbine, see Morin's Expériences sur les Roues Hydrodynamiques Appelées Turbines, 1838; Comptes Rendus, for July 1838; and the Notice on Vortex Water-Wheels in the British Association Reports for 1852.

(b.) Wheels with Passages.

496. Under wheels of this kind we may mention the Danaide of M. Manouri d'Ectot. The principal piece of this machine is a tun or small tub of white iron, having a hole pierced in its bottom, through which passes the axis of rotation, and by which the water escapes. The axis turns with the trough upon a pivot, and is fixed above to a collar.

A drum of tin plate, closed above and below, is fixed upon the axis of, and placed within the trough, so as to be concentric with it, and to leave only between the outer circumference of the drum and the inner circumference of the trough, an annular space not exceeding 1½ inches. This annular space communicates with a space less than 1½ inches, left between the bottom of the drum and the bottom of the trough, and divided into compartments by diaphragms fixed upon the bottom of the trough, and proceeding from the circumference to the central hole in the bottom of the trough.

The water comes from a reservoir above by one or two pipes, and makes its way into this annular space between the trough and drum. The bottom of these pipes corresponds with the level of the water in the trough, and they are directed horizontally, and as tangents to the mean circumference between that of the trough and of the drum. The velocity which the water has acquired by its fall along these pipes, makes the machine move round its axis, and this motion accelerates by degrees, till the velocity of the water in the space between the trough and drum equals that of the water from the reservoir; so that no sensible shock is perceived of the affluent water upon that which is contained in the machine.

This circular motion communicates to the water between the trough and drum a centrifugal force, in consequence of which it presses against the sides of the trough. This centrifugal force acts equally upon the water contained in the compartments at the bottom of the trough, but it acts less and less as this water approaches the centre.

The whole water, then, is animated by two opposite forces, viz., gravity, and the centrifugal force. The first tends to make the water run out at the hole at the bottom of the trough; the second, to drive the water from that hole.

To these two forces are joined a third, viz., friction, which acts here an important and singular part, as it promotes the efficacy of the machine, while in other machines it always diminishes that efficacy. Here, on the contrary, the effect would be nothing were it not for the friction, which acts as a tangent to the sides of the trough and drum.

By the combination of these three forces, there ought to result a more or less rapid flow from the hole at the bottom of the trough; and the less force the water has as it issues out, the more it will have employed in moving the machine, and of course in producing the useful effect for which it is destined. The moving power is the weight of the water running in multiplied by the height of the reservoir from which it flows above the bottom of the trough; and the useful effect is the same product diminished by half the force which the water retains when it issues out of the orifice below.

In order to ascertain, by direct experiment, the magnitude of this effect, MM. Prony and Carnot fixed a cord to the axis of the machine, which passing over a pulley, raised a weight by the motion of the machine. By this means, the effect was found to be \( \frac{1}{3} \)ths of the power, and often approached \( \frac{2}{3} \)ths without reckoning the friction of the pulleys, which has nothing to do with the machine. This effect exceeds that of the best overshot wheels. See the Report of the Institute, 23rd August 1813; or Thomson's Annals of Philosophy, vol. ii., p. 412.

M. Burdin, in 1833, invented a Danaide with axis vertical, which may be seen described in the Annales des Mines for 1836. D'Ectot's may be seen in the Journal des Mines, tom. 34.

(c) Reacting Machines, or Wheels of Recoil.

497. We have hitherto considered the mechanical effects of water as the impelling power of machinery, when it acts either by its impulse or by its gravity. The reaction of water may be employed to communicate motion to machinery; and it is believed that a given quantity of water, falling through a given height, will produce greater effects by its reaction than by its impulse or its weight.

Barker's Mill, or the Wheel of Recoil.

498. This machine, which is sometimes called Parent's mill, is represented in fig. 89, where MN is the canal that conveys the water into the upright tube TT, which communicates with the horizontal arm AB. The water will therefore descend through the upright tube into this arm, and will exert upon the inside of it a pressure proportioned to the height of the fall. But if two orifices, A and B, be perforated at the extremities of the arm, and on contrary sides, the pressure upon these orifices will be removed by the efflux of the water, and the unbalanced pressure upon the opposite sides of the arm will make the tube and the horizontal arm revolve upon the spindle D as an axis. This will be more easily understood, if we suppose the orifices to be shut up, and consider the pressure upon a circular inch of the arm opposite to the orifice, the orifice being of the same size. The pressure upon this circular inch will be equal to a cylinder of water whose base is one inch in diameter, and whose altitude is the height of the fall; and the same force is exerted upon the shut-up orifice. These two pressures, therefore, being equal and opposite, the arm A will remain at rest. But as soon as the orifice is opened, the water will issue with a velocity due to the height of the fall; the pressure upon the orifice will of consequence be removed; and as the pressure upon the circular inch opposite to the orifice still continues, the equilibrium will be destroyed, the arm will move in a retrograde direction.

499. The upright spindle D, on which the arm revolves, is fixed in the bottom of the arm, and screwed to it below by a nut. It is fixed to the upright tube by two cross bars, so as to move along with it. If a corn-mill is to be driven, the top of the spindle is fixed into the upper millstone m. The lower quiescent millstone n rests upon the floor K, in which is a hole, to let the meal pass into a trough below. The bridgepiece ab, which supports the millstone tube, &c., is moveable on a pin at a, and its other end is supported by an iron rod fixed into it, the top of the rod going through the fixed bracket O, furnished with a nut c. By screwing this nut, the millstone may be raised or lowered at pleasure. If any other kind of machinery is to be driven, the spindle D must be prolonged above the hopper H, and a small wheel fixed to its extremity, which will communicate its motion to any species of mechanism.

500. Barker's Mill, or Segner's Wheel, as it is sometimes called, has not been extensively used in moving machinery, owing chiefly to the lower pivot D being made to bear the whole weight of the water in the cylinder, besides the weight of the apparatus. The considerable resistance, however, arising from this cause has been ingeniously set aside by Althans of Sayn, who has so arranged the machine that the water enters the horizontal cylinder DT from below. Such a mill may be had by removing the apparatus HK, the horizontal cylinder AB being above, and NT below. But in such an arrangement, the upright cylinder TT is now stable, and AB only rotates round a pivot, and the friction also is very much reduced, the weight of the wheel, and all that is fastened to it, being entirely supported by the column of water. An advantage is derived by having a rotating cylinder similar to an S, placed thus \( \infty \); for water issuing from the orifices of a straight cylinder issues from it with a certain velocity, but in passing through a cylinder with curved arms, it gradually loses its impetus, till at last it falls from the orifices with no velocity whatever.

In several parts of Scotland these Barker's mills are known as Scotch Turbines.

(c) Albert Euler's Reacting Machine.

501. This machine was first described in the Mémoires de l'Académie de Berlin for the year 1754; it consists of two parts, GH the upper (fig. 90), and EF the lower. The upper portion is immovable, and forms a cylindrical and annular reservoir, receiving water within the annulus PP, from the canal R; at its lower part I, I, are a set of straight pipes, but inclined at an angle determined by calculation, so that the water may descend with proper obliquity into the lower vessel. These pipes are represented by the lines I i. The lower part is moveable round the common axis OO, and has also an annular trough EE, from the bottom of which, inside the vessel, pipes descend and discharge from a rectangular end F, the water which passes through them. The velocity with which the water issues is that due to the height EF; and a rotary and retrograde motion takes place in the lower half of the machine.

M. Burdin gives an account of a reacting turbine of his own invention, along with his experiments on it, in the Annales des Mines for 1828. CHAPTER II.—MACHINES HAVING AN ALTERNATE MOTION.

502. The second kind of wheels which we come to describe, are those having an alternate motion; of these, two only are known in the arts,—the Water-column Machine, and Montgolfier's Hydraulic Ram.

(a.) Water-column Machines.

503. The water-column machine consists of a cylinder or great body of a pump, in which a piston is driven backwards and forwards by the weight of a high column of water contained in an upright pipe. To the piston-end is adapted a working beam, which transmits a motion to the common pumps; sometimes also the up and down motion is transformed into a motion of rotation. The first machines of this kind were constructed in 1748 by Haël, and were known as Hell Machines. They were used at that time in the mines of Hungary, and in different parts of Germany; but were not very effective. It was only about 1817 that Reichenbach made them what they now are. One of these engines, as improved by him, was constructed at Illsang, which raised a column of salt water to a height of 1145 feet. One of the best of these machines is that which works at a depth of 410 feet below the surface in the mines of Huelgoat, in Bretagne, a description of which, by M. Junker the engineer, is given in the Annales des Mines, 1835; and in M. Theodore Fischer's Handbuch der Hydraulik, 1835.

(b.) Whitehurst's Machine, and the Hydraulic Ram.

504. Mr Whitehurst gives, in the Phil. Trans. 1775, an account of a machine for raising water to a considerable height by means of its momentum. This machine consisted of a reservoir AM, (fig. 91), having its surface M on a level with the bottom of another reservoir BN. The main AE is about 200 yards long; and F is 16 feet below MB. The valve-box D has a valve α, and into the air-vessel C are inserted the extremities m of the main pipe, bent downwards to prevent the air being driven out when the water is forced into it. Now, from the distance of F below MB, as soon as the cock F is opened, a column of water 200 yards long is put in motion; and if the cock F be suddenly shut, the water rushes through the valve α and condenses the air in C. Since this condensation of the air must take place every time that F is suddenly stopped, the air in C will react on the water and force it up into the reservoir BN, by the pipe CB.

505. The Hydraulic Ram of Montgolfier is based on this same principle. Although this engine is evidently an improvement on Whitehurst's Machine, yet Montgolfier afterwards claimed the entire merit of the invention.

The pipe A, or body of the Ram, conveys water from an elevated reservoir, not represented in fig. 92, to the working part of the machine, or head of the Ram. This head consists of the short pipe at B, opening upwards, where B is a valve connected with a kind of stirrup surmounting the opening. When B is at its highest, the water enters the air-vessel C, and is forced from it into an outer air-vessel F, whence it is expelled into the ascension pipe G to the required height.

Suppose now, that the water is moving along the pipe A, and that B is at its lowest; then as B is in its way, and as it has no passage except by the circular sides of B, the valve B will be thus gradually forced upwards, till at last the water cannot escape by this opening. The head of water is now forced along the pipe with an increased velocity, exerting a pressure on the walls, and escaping by the valves E, E, the only yielding parts; a volume will therefore be poured into the vessel F, which, by the condensation of the air, will be forced up the ascension pipe G. When the valves E, E open, the velocity of the water in A immediately diminishes, and after a very short time, these valves close, for the water has now lost all its impetus. B also, which is no longer pressed upward, gradually falls to its original position, and then prepares anew for a second stroke.

The pressure of the air in C plays a very important part in the machine. When the water is sharply arrested from escaping, by the valve B closing, it produces a violent shock, in virtue of which the valves E, E open so as to admit water into the vessel F. But the air in C meeting this shock, produces the greatest amount of work; and when, by the agency of the air in C, the water has lost all its velocity, this air, in virtue of its elasticity, repels the water back into the pipe A. The valves E, E will then shut as soon as the pressure of the air and water of F inwards on E, E exceeds that outward pressure of C on E, E. The backward motion also of the water, in consequence of its acquired velocity, continues even though the pressure has been reduced to that due to the height of the fall; that is to say, although the pressure in C is less than that of the atmosphere, the water continues to move backwards in the pipe A. It is owing to this interior sucking that the valve B falls to prepare for a second stroke.

The air again in the vessel F has also to play an important part. It is of use in maintaining a uniform and continuous motion in the ascension of the water by the pipe G. When the valves E, E open, the water entering, compresses the air in F, but is not itself immediately forced up the pipe G, as it would be were there no air in F. The suppression of the air, therefore, in F, would require a greater force to raise a high column of water from a state

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1 See the Journal des Mines, vol. xlii., No. 73. of rest, and the valves E, E would remain open a shorter time at each stroke of the ram; and, consequently, also, would throw in a smaller volume of water.

It is necessary, therefore, that the air in the vessels C and F be carefully supplied. Each volume of water, no doubt, brings along with it so much air; but as each is expelled, more air is carried out than is introduced. A small pipe H, then, having a valve opening inwards, and right under E, throws in a volume of air at the time when the backward motion of the water is taking place in pipe A, the pressure of the rarified air in C being less than that of the atmosphere. The vessel C being now full of air, then as soon as a new stroke of the ram takes place, the valves E, E open and introduce a volume of air into F. The water which passes over the valve B is conveyed away by the channel D.

CHAPTER III.—MACHINES FOR RAISING WATER.

507. The machines for raising water are—(a.) Pumps; (b.) Screw of Archimedes; and, (c.) Pail or Bucket Machines.

(a.) Pumps.

508. Pumps have for their object the raising of water to any height. A pump consists of a cylinder or body of the pump, in which an upward and downward movement takes place by means of a cork or piston, which is closely fitted to the walls of the cylinder. By the intervention of valves, a communication is established between the cylinder and the various pipes necessary for the throw of the pump. There are two pipes especially to be attended to; the lower is the sucking pipe, the upper is the forcing pipe.

Pumps are divided into three classes, and named according to the circumstances in which the piston acts,—(a.) Sucking Pumps. (b.) Forcing Pumps. (c.) Sucking and Forcing Pumps.

(a.) 1. The Sucking Pump.

509. The common sucking pump consists of the cylinder DCCD (fig. 93), and the suction pipe BAA.B of smaller diameter. These two parts are united by flanges E, F, tightened by bolts C, C. The level of the water in which the pump is plunged is ZY, and AA has a grating across it to prevent filth from entering the pump. OPM is a conical air-tight piston, and A' its rod; the mouth-piece, by which the water issues, may be at V or at X. The valves opening upwards are H and N; and R joins piston and rod. In order to explain the action of the common pump, suppose that the piston is at the top of the barrel, or at TS, then the top of the piston-rod is at its highest; let also the valves H and N be shut. Now, as the piston-rod A' descends, there is a tendency of the air within the barrel to be compressed; the valve N must necessarily fly open upward, whereby the air escapes. When the piston M has reached GE, or the bottom of the cylinder, the valves are closed and a volume of air has escaped by N equal to the capacity of the barrel. Were the valve H permanently fixed, and were the piston to rise, then a vacuum would be formed between H and the bottom of the piston; but as H is hinged, immediately on M rising, the elastic force of the air between H and YZ forces up that valve, and the cylinder will now be filled with rarified air from pipe B, when the piston is again at the top. Let the piston redescend, and that volume of air will also be expelled, and so on. But it is clear, that as more and more air is withdrawn, the pressure of the rarified air is less and less able to balance the atmospheric pressure on the surface YZ of the water, and the elastic force being proportional to the density, the water will rush up and fill the barrel.

Suppose now that the air is completely exhausted, and that M as well as the water is at H; then on M rising to D, the atmospheric pressure at YZ will force the water from the suction pipe into the barrel so as to fill it; but on M descending, since it cannot penetrate without displacing the water, the valve H must close, and N open, so that the whole volume of water which was below will now be above M. As the piston is again raised it will turn all the volume of water above it into the reservoir VX, so that it may be emptied by a mouth-piece in the side; the barrel is again filled, however, and must be discharged as before.

This is the action of the common suction pump. Before the time of Galileo, or rather before Torricelli showed experimentally the pressure of the atmosphere, it was believed that the water was sucked up in the pipe HA, owing to the abhorrence of nature to a vacuum. But as Torricelli showed that an atmospheric column, extending through the whole height of the atmosphere, would balance a column of water 32 feet high of the same sectional area, it follows that, in order to raise water by the common pump, the valve H must on no account be more than 32 feet above the surface YZ of the water. If it exceed that height no water will be raised.

With respect to the pressures which the faces of the piston experience, we see that on M descending, when the barrel is full, the pressure on the upper and lower face is not exactly the same, for the faces are not of the same depth, and the water is slightly resisted while passing through the valve H, which shows us that this opening should be as large as possible. As the difference, however, is excessively small, the piston may be regarded as having no resistance to overcome in passing from D to C. It is otherwise when the piston is rising. It then acts as a full piston, and has to support different pressures on its two faces. The upper face supports the atmospheric and the fluid column, while the lower face is only pressed by the atmosphere, less a fluid column extending from the bottom of the piston to the level YZ of the water. The difference of these pressures may be regarded as the pressure of a fluid column, having a section of the piston for its base, and its depth the vertical distance between the mouth-piece, say V and YZ. The descent and ascent of the piston constitute one stroke of the pump.

510. In the common pump, let A = area of piston; l = length of stroke; h = vertical height of the bottom of the barrel above the surface of the water to be raised; \( k = h + l \) for raising vertical height of the pipe by which the water is discharged from the water to be raised; \( w \) is the weight of a cubic foot of water; \( P \) is the work applied at each double stroke; \( m \) is the ratio of the work done to the work applied, or the relative efficiency of the pump. Then work in raising water into the barrel \( = A \cdot w \cdot (h + l) \), and work in raising water from the barrel \( = A \cdot w \cdot \frac{1}{2} l \).

\[ \text{Total work} = A \cdot w \cdot (h + \frac{1}{2} l) + A \cdot w \cdot \frac{1}{2} l = A \cdot w \cdot (h + l) \]

hence \( mP = A \cdot w \cdot (h + l) = A \cdot w \cdot k \), and \( l = \frac{mP}{A \cdot w} \).

### 2. Self's Pump, with Suction Air Chamber.

511. In the agricultural department of the Great Exhibition of 1851, Mr Self exhibited a pump with suction air chamber. The peculiarity in the construction of this pump consists in its having an air chamber connected with a common suction pump, the air chamber communicating with the suction pipe immediately below the barrel. When working the common pump, a sudden jerk given at the beginning of the stroke, sometimes separates the piston from the water in the barrel, causing a vacuum into which the air will find its way. Self's suction chamber is an arrangement calculated to remedy the defect, and save the labour accumulated in the water. Professor Mosely, in his report, says of this machine:—" It is immaterial in what proportions the work is distributed over the stroke, or under what varying degrees of pressure it is generated, provided that the pressure never exceeds that of the atmosphere on the surface of the piston. If this pressure be exceeded, the piston may separate itself from the water beneath it in the barrel, the pump drawing air; and this is more likely to occur at the commencement than at any other period of the stroke, the motion of the water at that point being necessarily slow.

To communicate a finite velocity to the water at the commencement of the stroke, or while the space described by the piston is still exceedingly small, requires a much greater pressure than afterwards; and the greater, as the section of the suction pipe is less as compared with that of the barrel, and as the lift is greater. Thus, at the commencement of the stroke, a finite velocity of the piston can only be obtained by an extraordinary effort of the motive power associated with the chance of drawing air and of a shock, if the pressure be suddenly applied. A remedy for some of these evils in the working of a pump has been sought in the application to it of a second air vessel, communicating with the suction pipe immediately below the barrel, or with the top of the suction pipe and the bottom of the barrel. The commencement of each stroke is eased by a supply of water from this air chamber to the space beneath it. The influx of the water into that space is aided by the pressure of the condensed air in the air chamber, and when the stroke is completed, the state of the condensation of this air is, by the momentum of the water in the suction pipe, restored, causing it to rush through the passage by which that pipe communicates with the air chamber. Thus, by this contrivance, the surplus work which remains in the water of the suction pipe at the conclusion of each stroke, is stored up in the compressed air of the air-chamber, and helps to begin the next stroke of the piston.

The nature of this action will be best understood from that of the hydraulic ram (505). The contrivance constitutes, indeed, in some respects, a union of the action of the ram with that of the pump; and, besides accomplishing the object for which it was applied, appears to have the effect of considerably economizing the power employed in working pumps.

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3. The Lifting Pump.

512. The object of the lifting pump is to raise water to a greater height than the common sucking pump can do. It is therefore nothing else than the common pump a little modified. ACDB (fig. 94) is the working barrel of the pump, with a side pipe AEGHP flanged, and terminating in a main or rising pipe IK, with a valve L. The top of the barrel is now shut by a strong plate MN; the piston rod Q works through the watertight part OP. The piston valve is R; the barrel and suction pipe are firmly bolted at D, C, and XV is the level of the water. The water first of all passes through the sucking valve T, opens the piston valve R, and fills the barrel BC; then when the piston rod is drawn up, the water is necessarily lifted and driven through the forcing valve L into the escape pipe IK.

The action of the lifting pump is very simple. Suppose that the piston R is at T, that all the valves are shut, and that the air in the barrel and suction pipe is highly rarified, then immediately on the piston rod Q rising, the suction valve T will be forced open by the upward pressure of the water, and the piston valve R will necessarily be closed, for it drives out the remaining air from the barrel; consequently the valve L will keep open. When the piston R has reached NM, the water has entirely filled the barrel, and the valve L is shut by the atmospheric pressure. Again, when the piston rod descends, the valve R opens and T shuts; and when R is again at T, having completed one stroke, all the water will be above the piston and fill the barrel. Hence, on R being a second time pulled up, it has now to lift a weight equal to a fluid column BC, together with the atmospheric column in pipe IK, for the valve L is pushed open; on R reaching AB, a volume of water has been sent into the pipe IK equal to that which filled the barrel. But the barrel BC is again filled, and when the piston R descends, the fluid column in IK shuts L, the piston valve R opens and T shuts. When the piston R rises a third time, it has now to lift the water in the barrel, together with that in the pipe IK; and hence it is clear that water may be raised to any height, provided only that the barrel be strong enough to sustain the weight of the superincumbent column in IK,—which must be raised before the valve L can open,—and the force necessary to work the common suction pump.

The distance of DC from XV must not exceed 32 feet, but IK may be of any length. The piston rod is always drawn upwards, and consequently the rod Q is subjected to a longitudinal strain. A slim rod will suffice for this purpose, since there is no compressing force. The whole force which has to be overcome by the working lever or handle of the machine is,—1st, the weight of a fluid column extending from the level XV to the level of water in pipe IK, the base being that of the piston, a weight which acts at the piston rod where it joins the short arm of the handle; and, 2d, the passive resistances caused by the motion of the water and of the solid walls of the pump.

Sometimes a small pipe (furnished with a cock) projects from the side of IK. When the pump is in action, and if

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1 Mosely's Report. the cock be turned, the water will escape by the small pipe, or raising so that in this condition the lifting pump becomes a common pump.

(b.) 1. The Forcing Pump.

513. The Forcing pump consists of a working barrel, ABCD (fig. 35), a suction pipe CDEF, and an ascension pipe, ONML. The part GHKI may be said to belong to the barrel; the second part IKLM is united to it, and the part LMNO is properly the ascension pipe. The cock or forcing valve is S, and RN is the suction valve. The object of the forcing pump is to raise water to a greater elevation than can be obtained by either of the former pumps. In the forcing pump the barrel AB is open to the atmosphere, the piston QVT is solid, and Q is a strong rod of iron. Suppose that the piston is being pushed down to R, then the valve S opens, and a volume of air equal to the capacity of the barrel is expelled by ON; let VT now rise, and R, which before was kept shut by the pressure, opens; S also shuts, and a volume of air from the suction pipe enters, and, expanding, fills the barrel. As the rarified air is more and more expelled, the water gradually rises in the suction pipe. Suppose that the piston VT and the water are at R, then, on VT rising, the water enters, and when VT reaches AB, the whole barrel is filled; the water has thus been sucked into the barrel. When VT begins to move down, R closes, and S opens; but it requires a forcing down of the piston to make it pass through the valve S into the pipe OL; the whole will be forced into this pipe as soon as VT is at R. A new volume will enter by R on the rising of VT, and S will be shut by the weight of the column in OL; an additional force, therefore, will be required to expel this second volume, owing to the column in OL.

A small force is required to lift the piston-rod, but the force requisite to overcome the resistance in pushing it down, is that which would balance a column of water of a height equal to the distance between the level of water in pipe OL and the bottom of the piston, the sectional area being that of the piston. Since a great pressure is necessary to force the water into the pipe OL, the rod Q must be strong to resist the compression.

514. In the case of the forcing pump, let \( h \) be the vertical height of the nozzle by which the water is discharged from the bottom of the barrel, then work produced by downward stroke \( = Alw(h - \frac{1}{2}l) \), where the quantities are the same, as in the former pump;

\[ \text{Total work} = Alw(h + \frac{1}{2}l) + Alw(h' - \frac{1}{2}l) \]

\[ = Alw(h + l) = P; \quad mP; \quad l = \frac{mP}{Alw(h + l)} \]

where \( m \) is the coefficient.

2. The Plunger Pole Pump.

515. It is very difficult at times to make the pistons move water-tight in their casings, especially where dirty water is to be pumped up, as in the drainage of mines, and in the case also of the Bramah press. If, therefore, pistons are to fit accurately, they must be solid, as in fig. 96, where O is a solid piston, nicely turned and polished, its diameter being a little less than that of the cylinder CAGD, and working through a water-tight collar or stuffing-box DC. The small cistern XY, around the head of the pump, is kept full of water, for the purpose of keeping moist the air-tight collar. The small pipe RSZ in this cistern, is that, when the plunger O is at M, water may be poured into it by the cock S, so that the barrel AD may just be full. The piston-rod works through four rings T, of soft leather; HIGA are bolts binding the parts together. Suppose, now, that the plunger-pole O is at M, that water has been admitted by RSZ, and that the cock S is shut, then, on the plunger being drawn up, the air from pipe B will force up M, expand, and fill the barrel, together with so much water, while N is kept shut by the pressure of the atmosphere. When O descends, it drives the water before it, forcing down M, opening N, and expels a volume of water and air by LK. But on O reaching M, then N closes. There is still so much air in the barrel, but a few strokes of the pump will expel it, so that afterwards water only fills the barrel.

One great advantage which this pump has, is that it does away with the necessity of boring and polishing large cylinders accurately true, which of itself was a task of extreme difficulty; and even though it be done, yet the constant rubbing of the piston on the sides of the cylinder would soon reduce the diameter of the piston, necessitating therefore its frequent removal and renewal. The only wear, however, in the case of the Plunger-pole pump, so called because it plunges into the barrel, is the rubbing of the pole on the leather casing, and also the piston-rod Y through T; but these collars can be replaced at little expense.

Although the air may generally be said to be expelled shortly after the pump has made several strokes, yet to secure the utmost effect of the machine, a small right-angular channel is cut in the pole, one end communicating with the cylinder, and the other at the top of the pole, where there is a cock. If any air should accumulate at the top of the barrel, the cock is opened, and it is forced out.

3. The Mining or Draining Pump.

516. In draining the water from a mine with a single forcing pump, it is clear that there would be a column of several hundred feet, and a piston-rod of the same length. The working of such a pump would evidently require to fulfil conditions which would be impossible. In order, therefore, to drain mines, the method sometimes adopted is to divide the whole length of the shaft into several equal parts, at each of which a pump is established. The principal rod extends from the top to the bottom of the mine, carrying at equable distances heavy plunger-pole pistons, which work up and down in the cylinders of their respective pumps, guided all at the same time by the upward and downward movement given to the principal rod by the steam-engine at the mouth of the mine. There are ascen- (c.) 1. The Fire Engine.

517. The pump used for extinguishing fires is a double forcing pump, and is called the Fire engine. The pipe by which the water is thrown upon the fire is very flexible, so that it may be directed at pleasure to any part of the conflagration, whilst the pump is acting. The height of the jet will thus be variable, and if the pipe be held in a horizontal position it will be nothing. But the object required in the working of the machine is not so much to send the water to the extremity of the flexible pipe, as to give the water a considerable velocity when the jet issues from it. A jet is thus produced of great range, which can easily be directed to distant parts, where the fire is to be arrested. In order to effect this, the jet which issues from the pipe should escape with a uniform velocity. Hence, for this purpose, we have on both sides of the machine a forcing pump, each moving alternately during the same time, and by their united action constituting a pump of double effect. The pistons S, R, of the pumps WX, TU (fig. 97), move at the same time, but in contrary directions; when one is up the other is down, and vice versa. The water introduces itself into the interior of the pumps by the valves O, G; when a pump is full, the piston descending, shuts G and opens K, and the same with the other; the water is thus thrown into the reservoir abcd, into which is plunged the ascension pipe fe, forming part of the flexible pipe. Although there are two forcing pumps employed for throwing water into the reservoir, yet this arrangement does not ensure that regularity of velocity of escape which might be expected, since a marked irregularity or a retardation of the water takes place every time that each piston changes the direction of its motion. The velocity of the water, however, is regulated by a volume of air lodged above YZ in the top of the air reservoir ae. This air is completely inclosed; it puts itself in equilibrium with the pressure of water under it; and its elastic force, acting on the surface of the water YZ, forces the water up the pipe fe to the stop-cock eg, whence it passes to tube h attached to the leatheren tube or hose. By means of the air vessel, then, the irregularities which the volume of water entering by I, K presents, are mainly felt in the reservoir ae into which they open, and are removed by the oscillations of the water which rise and fall alternately. It results from this, that the variations are very small, yet sensible in the velocity with which the water spouts out at the extremity of the hose. The long pipe or hose is of considerable transverse dimensions, but contracted to a narrow point at its nozzle. The water passes gently from f along the hose, and it is only on entering the narrow portion that it receives a considerable velocity of escape. The height, then, to which the water will be thrown depends on the condensation of the air in the air vessel, and on the elevation of the extremity of the pipe above the level of the water in f.

The fire-engine is made long and narrow; a section of its breadth is ABCD in fig. 97; the principal use of the length, is to support a frame-work which is balanced on a horizontal axis, of which g is the projection, parallel to the length, and right over the reservoir ae. At each end of this axis is a lever αβ, united at α and β by two long handles, of which the projections are α and β. Several men, standing on both sides of the engine, grip the handles so as to work the pump's pistons R, S, to which they are united by vertical rods. The long body of the engine serves likewise as a cistern into which pails of water may be poured, if there be not a convenient supply from a pipe F communicating with the cistern of the engine and the town mains.

2. Delahire's Pump.

518. Delahire's pump consists of a working barrel AB, shut at both ends A and B (fig. 98). C is the piston, and its rod O passes through a leather collar A. Communicating with the barrel, by the passages m and n which are near the top and bottom of the barrel, are two pipes H and K. The valves F and G open upward, D and E are similarly circumstanced. The action of the pump is evident. Suppose that the air has been withdrawn, that the water is at D, and the piston C close to n, when the piston-rod O is drawn up, all the valves will be shut except F and D, the latter of which must open first by the pressure of the water; the barrel, therefore, will be full when C is at A. If C descends, a volume of water must be expelled by E, which immediately opens on the descent of C, while D shuts. But as C is forced down to expel the water under it, the valve F instantly opens by the force of the water, and fills up the vacant space above C, the valve G falling by the pressure of the atmosphere. Now, the first volume of water was expelled by the pipe H, and it is just rising in that pipe while C has reached n; but as soon as C begins to rise again, the valve G is forced open by the water above C, and F shuts; the valve E is also closed by the fluid column above it, and D opens to fill the barrel under C. Hence we see that two barrels of water are expelled by the common pipe L at each stroke of the pump, so that the stream is continuous, or nearly so. There is, however, an awkward jar or shock at each change of the action; but this is entirely removed by 3. The Bramah Press.

519. The principle on which the Bramah press depends for its efficiency, is the equable distribution of pressure in fluids. Fig. 99 represents the machine, while fig. 100 is a section of it. AA is a very strong cylinder inclosing a plunger piston B, having on its top a plate C. The piston, wrought by means of water, rises with its plate, and presses with great force any substance upon it against the roof plate D firmly bound to the pillars EE. The water is introduced into the cylinder A by means of the injecting pump F. This injecting or plunger pump is worked by a lever GH, of the second kind, moving round H, the power being applied at the handle G. As the handle receives an upward and downward movement, the plunger piston B ascends and descends, and transmits pressure against the bottom of the piston B by means of the water. The movement of the piston I is guided by the circular hole K, in which the free end of the piston-rod moves; the piston-rod of I is connected with the lever GH. At every stroke, then, of this piston, water is sucked up by the valve M from a reservoir placed under this pump, and is driven into the pipe L communicating with the cylinder A. M and N are two valves, which establish and intercept alternately the communication of the body of the pump A with the sucking pipe at M, and the connecting pipe L. The forcing valve is N, and the sucker valve M.

It is of the utmost importance that the surface of the pistons with the walls of the respective cylinders should be as close as possible, especially at the tops of the cylinder, and that the packing between them should be perfectly water-tight; for if there be the smallest leakage, a volume of water would inevitably escape. A leak, however, near or about I, is of less moment than at some part of A. For were the motion of the injecting piston I somewhat rapid, all the water would not have time to pass out by the leak at A, and for raising therefore a portion of the water would always be effective. Special attention is always paid by engineers to the secure packing of the collars of the piston A, so that the pressure may be transmitted from one cylinder to another unchanged. On every descent of the piston I, a volume of water is forced through the valve at N. This valve closes the instant the piston has made its complete downward motion, and prevents the water from returning. The water thus thrown in, must be either compressed or must occupy a larger space. Practically speaking, water may be regarded as incompressible, and thus the water forced into the cylinder A must increase at every stroke, and the cylinder being too strong to yield, the piston B with its loaded plate must ascend. The piston will thus rise through a small space for every additional volume of water thrown in. As soon as the required effect has been produced, the pressure is immediately relieved by opening the discharge valve R, when the water returns to the cistern; and the piston B descends by its own gravity.

The packing of the piston B is too ingenious and important a matter to be passed over without notice. Let fig. 101 be a section of part of the main cylinder and piston, where A is the cylinder, KB the piston. KB is about 9 inches, DE is a recess in the former, about one-half inch deep; the collar FG is a double leather turned over a metal ring H, placed in this recess. Bramah, in order to prepare the collar, took an annular piece of leather and steeped it in a solution for some time. Having softened the leather, he next placed it over the top of the metal ring H. Now, when the water is pumped into the cylinder, it enters the narrow space between the walls of the cylinder A and piston B, then into the recess DE, and acting on the under surfaces of the turned-over leather on both sides of the ring H, forces the faces on the sides of the piston and on the back of the recess. The more water then that is pumped in, and as the piston begins to rise, the pressure on the inner leather face is increased, so that the piston and cylinder being perfectly air-tight by means of the firmly pressed leather, no water can insinuate itself between the edges F, G, and escape. The greater the pressure on the leather, the tighter will the packing become, provided only that the leathers be sound. When the machine is begun to be worked, the leather gradually rises up to the top of the recess by the mere pressure of the water, and even before the piston has commenced rising. The leather packing will thus assume the form as in fig. 102. As soon as the discharge valve R is open, so that the pressure may be removed, the leather is free and no longer pressed, and both it and the ring will slide down to the bottom of the recess. The edges of the leather are not so low as the lower edge of the ring H, in order to prevent the leather edge at F from coming between the cylinder and piston.

520. A similar contrivance renders the injecting pump water-tight. Fig. 103 represents the barrel mm, and n the plunger piston; aa is a copper ring lying horizontally between leather packing, which are the dark pieces. The ring has its upper and under surface scored, so that on the plunger being put in, and the piece bb screwed down, having a hole pierced in its mass to allow the working of the cylinder, the ring and the leather are in close contact, and the pressure squeezes out the leather on the side next the piston, and on the side next the barrel, thus preventing the water from escaping. The barrel is slightly curved as at cc, so that the water may get behind the leather, Machines press it against the side of the piston, and allow no escape between the piston and the leather.

When the pressure is more than three tons on the circular inch, the leathers wear very fast, and new ones must be put in.

When the machine begins to act, so as to compress a body between the roof and the plate, the pressure exercised is feeble; but as the body is more and more pressed, the resistance becomes so great, that a large power is required to overcome it. In such a case we remove the fulcrum of the lever GH from H to H', by the insertion of a pin, the bolt at H being afterwards removed. If HH' be half the former arm, then the former power will press the body with a greater force than before, and hence, also, a greater force will be transmitted through the water.

In order that the pressure transmitted by the water may not be too severe for the several parts of the machine, a safety valve P (fig. 104), is introduced near the pump F. A conical valve O intercepts a channel, by which the water may pass out with a force sufficient to overcome the weight P at the extremity of the lever of the valve, when the pressure becomes too great, i.e., when the pressure exceeds the limit beyond which we do not want it to pass.

A screw R, the lower extremity of which forms a valve, closes effectually another channel, by which the interior water might be prevented from communicating with the water cistern, during the working of the machine. When the action of the press is to be stopped, the screw R is turned in a convenient direction, and a communication is opened with the water cistern, into which the water flows from the cylinder AA, the piston of which, now no longer subject to pressure, gradually falls within its cylinder. This valve, connected with R, is called the discharge valve.

521. Given the force applied at the injecting pump-handle, to determine the pressure on the base of the working cylinder.

Let S be the area of the working piston's base, s that of the plunger pump, P the force applied at the injecting pump-handle, L the arm by which P acts at G, and l that by which acts Q, the resistance to be overcome by the injecting piston. Then, as the lever is one of the second kind, Q, l = P, L; ∴ Q = \frac{P \cdot L}{l} = resistance to be overcome. But the pressure exerted is proportional to the area pressed; and whatever be the pressure on a unit of surface of the injecting piston's base, the same pressure is transmitted to every other such area on the bottom of the working piston. Hence, action on base of B : action on base of I = base surface of B : base surface of I = S : s;

∴ Action on base of B = \frac{S}{s} \times action on base of I

= \frac{S}{s} \cdot \frac{P \cdot L}{l}.

This expression is equivalent to the pressure to which any substance placed between the plates C and D is subject; it is the same also as the weight raised, which call W;

∴ W = P \cdot \frac{L}{l} \cdot \frac{S}{s}.

But the areas of circles are as the squares of their radii, hence \frac{S}{s} = \frac{R^2}{r^2}, where R and r are radii of the base of the pistons B and I respectively;

∴ W = P \cdot \frac{L}{l} \cdot \frac{R^2}{r^2}.

Therefore, the efficiency of the press will be increased, by increasing either of the quantities P, \frac{L}{l}, or \frac{R}{r}.

Since the value of the ratio \frac{R}{r} may be increased to an almost indefinite extent, it is clear that the press may be said to have no limit to the pressure exerted, except the strength of the materials. But, in a practical point of view, the indefinite value of \frac{R}{r} may be said to be inapplicable; for when \frac{R}{r} is in value very large, the volume of water introduced at every stroke of the pump would be so small that the piston would ascend too slow for all practical purposes.

As it is desirable to save time in working with this press, and as one workman can easily work the instrument at the commencement, there are generally two injecting pumps, a large and a small one, standing side by side, and both communicating with the forcing valve. If the diameter of the piston of the large injecting pump be double that of the smaller, and their barrels of the same length, then the volume of water forced in at every stroke by the former will be four times greater than that of the latter. Hence the first pump will raise, at each stroke, the working piston through any space quicker by four times than that of the second piston. Again, since the pressure is proportional to the area, a man at the large injecting pump would only require to exert a force one-fourth that which he would require to employ at the small injecting pump, supposing their levers to be of the same length. Therefore, for the sake of speed, the large pump is worked first; and, when the effort required becomes too great, the small one. The effect evidently is the same as the shifting of the fulcrum point of the lever.

Further, in order to guard against accident, as, e.g., if the forcing valve N should be prevented from working by the intervention of dirt sucked up by M when the plunger I has made its descent, a hole is drilled in the bottom of the piston I, and communicating with another on the side at a small distance above, as represented by the dotted line (fig. 103). Now when the pump is worked properly, the horizontal orifice never rises above the leathers; but, if the plunger be raised a little higher, then the water will find its way between the metal surfaces of the plunger and barrel, and immediately the pressure on the base of the working piston will be diminished.

522. The Bramah press is perhaps the most perfect hydraulic machine with which we are acquainted. It is subject to little or no friction, nor to the rapid wear of its metal surfaces in contact; the only repairs that it may require are the renovation of the leathers, and yet these, under ordinary pressures, will last several years. This hydraulic press is used in almost every department of industry. It is used for throwing light articles into small bulk; for extracting oil from hides previous to tanning; for pressing cloth; for extracting moisture from paper. It is used in the manufacture of gunpowder, sugar, wax-candles, vermicelli, &c. It is peculiarly adapted for testing the strength of cables and masses of metal; for extracting old piles, and uprooting trees; for raising or sustaining a building that has sunk a little; and also for quarrying purposes. The printer and chemist know its value; and recently the engineer successfully accomplished, by means of the Bramah press, the elevation of the Menai Tubular Bridge.

4. The Centrifugal Pump.

523. Mechanicians have endeavoured to replace the up- ward and downward action of the piston in machines for raising water into a rotatory, or a motion always in the same direction. Bramah, in England, and Dietz in France, attempted to produce a machine which would work in this manner. That which the latter invented is represented in fig. 105.

Machines of this description are known as Centrifugal pumps; attention has been directed within these few years past to these engines on account of their great efficiency. The principle of their action will be understood from the following description. A cylinder AA (fig. 105), revolves within a fixed one PQ, in the annular space BB, each having the same axis. In the rotating cylinder are four plates or pieces C, C, C, C, which extend nearly, but not quite to the fixed cylinder; these plates are inserted in corresponding openings on the exterior surface of the rotating cylinder, and the space BB is divided into four compartments, each separated from the other. The exterior and interior edges of the plates are not parts of the exterior and interior surface of AA, although they move with it; these edges, which are contiguous from each other, approach when towards the right the centre O of the ring AA, such that the exterior edge of each plate in turn comes to form part of the outer surface of AA, while in other parts they form part of its inner surface. It is the form of the space BB which causes the plates C, C to glide into the slots of AA, and to near and go from alternately the axis of rotation. Hence, the four compartments are not always of the same capacity; their magnitude increases as the plates are distant from, but decreases as they approach, the axis of rotation. There are two openings in the fixed cylinder, each protected by a bent surface or opening valve, attached at one end to the fixed cylinder, while the other end is free; the one communicates with the suction pipe M, the other with the forcing pipe N. As the cylinder AA turns, one compartment is presented to M; it increases its capacity, sucks in water from that pipe, and becomes filled at the moment of its greatest dimension. This space diminishes on whirling round to the opening of the forcing pipe N, and a centrifugal force being imparted to it, drives the water forward, and compels it to discharge its contents into that pipe, and so on with all the others. We see, then, that this centrifugal pump acts at once as a sucking and a forcing pump; and fulfils, moreover, the conditions of a pump of double effect, since the movement given to the water in both pipes is continuous.

When the pump is set in motion the air from M is sucked out, and being carried round empties itself into N, so that the machine acts also as an air pump.

624. Mr Appold exhibited at the Crystal Palace in 1851 the form of a centrifugal pump, for which he was adjudicated the council medal. In Appold's pump the water enters by a circular opening, 6 inches wide, surrounding the axis with which the fan is connected. The fan-blades are 3 inches wide, and revolve within a fixed cylinder 12 inches diameter. The fan-blades consist of six curved arms. These curved fins are so disposed that the extremities at the fixed cylinder move tangentially, while the other extremities are fixed on to the rotating cylinder; the blades themselves pass through a stuffing-box in the side of the casing, working between two circular cheeks, and running close to them, but without actual contact. The outer revolving surfaces are thus shielded from the water, to which a free ingress is given, and a large space is left all round the circumference of the fan, to facilitate the escape of the discharged water.

Mr Appold showed the decided advantage of curved fins over straight ones. When the curved fins were used, the jury found, from the experiments, that with a lift of 19½ feet, and a velocity of 788 revolutions a minute, and a circumferential velocity of 2476 feet per minute, the discharge was 1236 gallons a minute, and the useful effect was 0·68 of the power. With nearly the same lift, a fan having straight inclined arms, and rotating at the rate of 690 revolutions a minute, and a circumferential velocity of 2168 feet in the same time, the discharge was 736 gallons; the useful effect was only 0·43 of the power. With straight radial arms, and nearly the same velocity, the discharge was 474 gallons, and the useful effect only 0·24 of the power.

But Appold's machine is supposed to have a useful effect of 0·72 of the power, with a circumferential velocity of 1627 feet a minute, precautions being adopted which could not be employed in the experiments made at the Great Exhibition. The largest pump of this kind as yet constructed, is that at Whitlesea Mere for draining purposes. The wheel is 4½ feet in diameter, and its average velocity is 90 revolutions, or 1250 feet per minute; it is driven by steam with 40 lb. on the square inch, and raises 15,000 gallons to a height of 4 or 5 feet in one minute. The centrifugal pump is most advantageous for low lifts, especially for those below 20 feet. Its best application is as a tidal pump, where the height of the lift is continually varying.

625. In the Centrifugal pump we have the same principle as will be seen in Barker's mill; the object of each, however, is different. In the latter, a fall of water through a hollow upright cylinder, with an axis passing through it, gives a rotatory motion to that vertical axis; while in the former, a rotatory motion is given to a vertical axis, in order to raise a column of water. Let P = the work in one second, applied in giving motion to the vertical axis; h = height to which the machine raises a column; v = absolute velocity of water after discharge; V = velocity due to height; P' = useful work per second; then, as in the reaction wheel, work done at efflux = work due to the centrifugal force, less the work spent in raising the water; hence \( v = \sqrt{V^2 - 2gh} \) = velocity of water due to the pressure in the vertical pipe, as well as that which is due to the centrifugal force; m = the weight of a volume discharged in one second; \( e = \sqrt{V^2 - 2gh} - V \). So also the work applied = work in raising the water + work in water after efflux;

\[ P = mgh + \frac{e^2}{2g} = \frac{m}{g}(V - \sqrt{V^2 - 2gh})V, \]

which, by expansion, gives, when V is infinitely increased, \( P = mgh \). The effective nature of the machine will be expressed by

\[ Q = \frac{P'}{P} = \frac{gh}{(V - \sqrt{V^2 - 2gh})V}. \]

The maximum effect of the machine, or Q, will take place when V = infinity.

Professor Mosely, in his Report, observes, respecting Mr Appold's centrifugal pump, that if the vanes be straight, it is evident that whatever may be the velocity of the water in the direction of a radius, when it leaves the wheel, its velocity in the direction of a tangent will be that of the circumference of the wheel, so that the greater the velocity of the wheel, the greater will be the amount of vis viva remaining in the water when discharged, and the greater the amount of power uselessly expended to create that vis viva. If, however, the vanes be curved backwards as regards the motion of the wheel, so as to have nearly the direction of a tangent to the circumference of the wheel at the points where they intersect it, then the velocity due to the centrifugal force of the water carrying it over the surface of the vein in the opposite direction to that in which the wheel is moving, and nearly in the direction of a tangent to the circumference, will, if this velocity of the water over the vane in the one direction be equal to that in which the vane is itself moving in the other, produce a state of absolute rest in the water, and entire exhaustion of vis viva. And in whatever degree the equality of these two motions—of the water in one direction over the vane, and of the vane itself in the opposite direction—is attained, in that same degree will the water be delivered in a state approaching to one of rest. The expedient of curved vanes is adopted in Mr Appold's pump.

With regard to the admission of water to the wheel, it is obvious that it should pass directly from the suction-pipe into the wheel without the intervention of any reservoir in which the vis viva of the influent stream—communicated in the act of rising through the pipe—may expend itself; and that such space should be allowed at the centre as not to alter the dimensions of the influent stream. It would further seem expedient, by means of properly constructed channels, to divide the water into separate streams, and to give to these divergent streams such curvatures as would facilitate their entrance upon the channels formed by the vanes, as in the Turbine or Reaction Wheels.

It is obvious that the tendency of the centrifugal force continually to increase the velocity of the water over the vanes as it recedes from the centre, cannot take effect in respect to all the particles of water in the same section, unless the sections of the channels diminish. If they do not, some of the particles of water in each section must be continually retarded, and power be uselessly expended in producing this retardation; whilst the current cannot but suffer from it a disturbance destructive of its vis viva.

This diminution of the sections of the channels might probably best be effected by giving to the sides of the wheel the forms of conical disks; an expedient which is adopted in Mr Lloyd's blowing machines, and in Mr Bessemer's centrifugal pump.

The communication of motion to the water of the reservoir in which the wheel revolves, and into which the water is discharged, should by every practicable expedient be avoided; and for this object the water should be kept as much as possible from the sides of the wheel. This is effected in Mr Appold's pump by fixing the wheel between two cheeks which project from opposite sides of the reservoir. The velocity with which the wheel must be driven depends upon the height to which the water is to be raised. Beyond a certain height this velocity is practically unattainable. But long before this limit is reached, it becomes inconsistent with an economical application of the power which drives the pump. It is probably therefore only in comparatively small lifts, where a large quantity of water is to be discharged, that the centrifugal pump will be found useful.

The Screw-Engine of Archimedes.

526. The screw-engine invented by Archimedes is represented in fig. 106, where AB is a cylinder with a flexible pipe, CEHOGF, wrapped round its circumference like a screw. The cylinder is inclined to the horizon, and supported at one extremity by the bent pillar IR, while its other extremity, furnished with a pivot, is immersed in the lower orifice of the flexible pipe is raised to the top, and discharged at D. On some occasions, when the water to be raised moves with a considerable velocity, the engine is put in motion by a number of float-boards fixed at L, and impelled by the current; and if the water is to be raised to a great height, another cylinder is immersed in the vessel D, which receives the water from the first cylinder, and is driven by a pinion fixed at L. In this way, by having a succession of screw-engines, and a succession of reservoirs, water may be raised to any altitude. An engine of this kind is described in Ferguson's Lectures, vol. ii., p. 113.

527. In order to explain the reason why the water rises in the spiral tube, let AB (fig. 107) be a section of the engine, BCdDE the spiral tube, BF a horizontal line or the surface of the stagnant water which is to be raised, and ABF the angle which the axis of the cylinder makes with the horizon. Then, the water which enters the extremity B of the spiral tube will descend to C, and remain there as long as the cylinder is at rest.

But if a motion of rotation be communicated to the cylinder, so that the lowest part C of the spiral BCD moves towards B, and the points d, D, E towards C, and become successively the lowest parts of the spiral, the water must occupy successively the points d, D, E, and therefore rise in the tube; or, which is the same thing, when the point C moves to c, the point d will be at C; and as the water at C cannot rise along with the point C to c, on account of the inclination of Ce to the horizon, it must occupy the point d of the spiral, when C has moved to c; that is, the water has a tendency to occupy the lower parts of the spiral, and the rotatory motion withdraws this part of the spiral from the water, and causes it to ascend to the top of the tube. By wrapping a cord round a cylinder, and inclining it to the horizon, so that the angle ABC may be greater than the angle ABF, and then making it revolve upon its axis, the preceding remarks will be clearly illustrated—If the direction of the spiral BC should be horizontal, that is, if it should coincide with the line BF, the water will have no tendency to move towards C, and therefore cannot be raised in the tube. For a similar reason, it will not rise when the point C is above... the horizontal line BF. Consequently, in the construction of this engine, the angle ABC, which the spiral forms with the side of the cylinder, must always be greater than the angle ABF, at which the cylinder is inclined to the horizon. In practice, the angle of inclination ABF should generally be about 50°, and the angle ABC about 65°.

528. The screw of Archimedes is now generally constructed as shown in the annexed figure, where AB is the axis of the screw, having a flat plate of wood or thin iron coiled, as it were, round the axis, like a spiral, or the threads of a screw. The plane of this plate is perpendicular to the surface of the cylindrical axis AB, but is inclined to the direction of the axis at an angle which must always be greater than the angle which the axis AB forms with the horizon when in use. This spiral plate, which is nothing more than a wooden screw with a very deep and narrow thread, is fixed in a cylindrical box CDEF, so as to form a spiral groove, as it were running up the tube from B to A, which is exactly the same thing as if a pipe of lead or leather had been wrapped round the cylindrical axis, as in fig. 108. If the outer case CDEF is fixed so that the screw revolves within it, the engine is called a water screw-engine. In the common screw-engine, Eytelwein has shown that the screw should be placed in such a manner that only one-half of a convolution may be filled at each revolution. When this condition, however, cannot be fulfilled, from the height of the water being variable, he gives a preference to the water screw, notwithstanding that in this case one-third of the water generally runs back, and the screw is apt to become clogged by impurities or weeds.

529. In a screw-engine erected at the Hurlet Alum Works, for raising the alum liquor, the length of the screw is 127 feet, its inclination to the horizon 37° 36′; the height to which it raises the liquor 76 feet 9 inches, the octagonal axis of the screw 8 inches in diameter, the diameter of the spiral 22 inches, the thickness of the covering 2 inches, the distance of the threads 9 inches, the number of the threads 168, the thickness of the spiral 2 inches, the width and depth of the spiral tube 7 inches each. The screw is sustained upon five sets of pivots or rollers, each set consisting of two rollers. The engine is driven by a water-wheel, which performs one revolution while the screw performs two. The quantity of liquor raised is 70 wine gallons; and as its specific gravity is 1.065, the quantity discharged in an hour is 17 tons. The screw is built wholly of wood, as the alum liquor acts upon iron.

530. The theory of this engine is treated at great length by Hennert, in his Dissertations sur la vis d'Archimède, Berlin, 1767; by Pitot, in the Memoirs of the French Academy; and by Euler, in the Nov. Comment. Petrop., tom. v. An account of Pitot's investigations may be seen in Gregory's Mechanics, vol. ii., p. 348. See also Eytelwein's Handbuch der Mechanick, ch. xxi.; and Journal des Mines, tom. xxxviii., p. 321.

531. The Zurich machine consists of a cylindrical or spiral pipe, having one extremity dipping into the water to be elevated, and its other end communicating with an ascension pipe, whereby the water is discharged. This machine was devised, in 1746, by an ingenious pewterer of Zurich, named Wirtz, and its theory has been investigated by Bernoulli, Eytelwein, Robison, and others. When the machine was first erected, and its capabilities tested, it was used successfully at Florence and in Russia. It is now scarcely to be found out of Holland, where a considerable number are to be seen driven by means of windmills. Fig. 109 will give us an idea of this machine and its manner of working. The dipping extremity of the pipe is scooped like a spoon or horn, and capable of taking in a volume of water which will fill half a coil.

532. After the scoop has emerged, the water passes along the spiral by the motion of it round the axis, and drives the air before it into the rising-pipe, where it escapes. In the meantime, air comes in at the mouth of the scoop; and when the scoop again dips into the water, it again takes in a similar quantity. Thus there is now a part filled with water and a part filled with air. Continuing this motion, we shall receive a second round of water and another of air. The water in any turn of the spiral will have its two ends on a level; and the air between the successive columns of water will be in its natural state; for since the passage into the rising pipe or main is open, there is nothing to force the water and air into any other position. But since the spires gradually diminish in their length, it is plain that the column of water will gradually occupy more and more of the circumference of each. At last it will occupy a complete turn of some spiral that is near the centre; and when sent farther in, by the continuance of the motion, some of it will run back over the top of the succeeding spiral. Therefore it will push the water backwards, and raise its other end, so that it also will run over backwards before the next turn be completed. And this change of disposition will at last reach the first or outermost spiral, and some water will run over into the horn and scoop, and finally into the cistern.

533. But as soon as water gets into the rising pipe, and rises a little in it, it stops the escape of the air when the next scoop of water is taken in. Here are now two columns of water acting against each other by hydrostatic pressure and the intervening column of air. They must compress the air between them, and the water and air columns will now be unequal. This will have a general tendency to keep the whole water back, and cause it to be higher on the left or rising side of each spire than on the right descending side. The excess of height will be just such as produces the compression of the air between that and the preceding column of water. This will go on increasing as the water

536. (γ) If instead of disks in the chapelet we substitute buckets or pails, each connected with an endless rope, then water may be raised to any elevation. Every time that a bucket enters the water it emerges full, and when it has reached its highest turning point, the bucket is emptied into the reservoir. This is the Noria or Bucket Machine. These machines are commonly used for draining and irrigating purposes.

537. (δ) If we conceive the wheel of art. 466 to be close to the circular course and turning in the opposite direction, and if we suppose the lower water-course the current of supply, then we shall have water raised to a height not exceeding half the diameter of the wheel. The machine is driven by a windmill or by steam, through the intervention of a toothed wheel acting on the teeth of the inner circumference of the wheel. By this arrangement the axis of the palette wheel is not much charged by the weight of water elevated, and consequently the friction of the wheel is much reduced. This is the Elevating Palette Wheel.

It is evident that this wheel is nothing else than an undershot or breast wheel, acting opposite to their mode of working.

538. (ε) The Elevating Bucket Wheel may be regarded as an overshot wheel working in the opposite direction. The buckets, or water compartments, begin to empty themselves by holes in the inner circumference of the wheel when near their highest turning point. All the water is received ere the wheel has passed its highest point.

539. (ζ) The Tympanum is also an elevating wheel, but much more efficient than either of the two former. It consists of a hollow circular drum, in which are four or more curved or spiral partitions, united to as many rectangular beams of a small height, of the same breadth as the drum, and proceeding from a revolving axis; the outer ends of these partitions are tangents to the drum. A crown wheel passes round the external circumference of the wheel, while a small crown wheel, turned by an engine, works into the teeth of the former wheel, and sets the whole apparatus in motion. Hence, as the mouths of the passages dip into the water a volume enters, which is carried by the motion of the wheel to seek the lowest point of the passage, which is at the axis of the wheel, where, finding a small opening, it empties itself into the reservoir. The friction of this wheel is inconsiderable.

These three last machines are much used in France for irrigating purposes.

CHAPTER IV.—ON THE BLOWING MACHINE OR WATER-BELLOWS.

The Trompe, or Water-Bellones.

540. Professor Magnus, at the close of his experiments on the motion of fluids, already detailed, gives an account of this piece of mechanism, the principle of which he thinks is to be referred to the penetration of a fluid by air. It is the so-called trompe, or water-bellows, which, according to Grignon, was discovered in Italy about the year 1640. The instrument is represented in fig. 109, drawn from the description given by Richard in his Études sur l'Art d'extraire immédiatement le Fer de ses Minéraux, p. 169.

B is a water-holder, kept always full by means of the canal Z. A, A are two tubes or hollowed beams about thirteen feet long, one of which is shown in the section. CC is a wooden cistern or barrel rendered air-tight; the tubes A, A widen toward the top; two boards p, p, inclined towards each other, are introduced above into each tube, thus forming a funnel-shaped narrowing of the orifice; the boards are kept asunder by the pieces of wood t, t. Below these pieces a number of apertures e, e, are made in the tube, through which air can enter. Similar apertures e', e' were made in Richard's instrument about half way up the tubes; from these water sometimes escapes, as Richard himself has found, and for this reason it would be better to omit them altogether. When the conical stopper of the funnel-shaped orifice p, p is drawn upwards, the water falls through the tube, and at the same time air is sucked in at the orifices e, e; as this is carried downwards by the water into the cistern, the air in the latter increases, and passing into the tube H, streams out of the opening b. At q, in the bottom of CC, an opening is made through which the descending water can escape; the magnitude of the opening is so arranged that the water can never sink down to it. A second trough is generally placed before the opening, in which the water must first ascend to escape over the edge.

Instead of air being sucked in by ee, we may have, in place of the boards p, p, two wooden funnels high enough to reach over the surface of the water. The water flows through the space between the funnels; this causes the fluid within them to sink and the air to enter. 541. The water-bellows is used in some departments of the south of France, and more immediately in the extraction of iron from its ores. In the year 1838 it was employed, with one or two exceptions, in the Department de l'Ariège.

542. Much has been written about the water-bellows; but the real cause of the descent of the air into the machine has never been properly explained, and it is yet totally unknown. Just as asserted last century that the water is changed by the violent motion into air! Venturi again referred the origin of the bubbles to the lateral motion imparted to the air by the moving stream. This would imply that a force of attraction exists between the air and water, sufficient to carry the former far beneath the surface, which is scarcely conceivable. One of Professor Magnus' experiments refutes this notion (422).

543. In order that the phenomena might be observed with the utmost accuracy, Professor Magnus constructed a model glass water-bellows (fig. 111), where N is the water-holder, in which, by means of a cork, the tube ab, 6 inches long, is fastened. The lower end of ab dips above into the tube cd, which is 6-5 feet long, and has an inner diameter of 4ths of an inch; at d it pierces a cork which closes the bottle AB, and it ends at g about two inches above the bottom of the bottle. Through the cork d two other tubes are introduced—de which can be closed by the cock e, and hik which serves as a manometer, and is filled with mercury to k.

544. When the tube ab had an opening at its lower end of 0-4 of an inch in diameter, as the water flowed through it downwards, a considerable portion of air was carried along with it into the bottle AB. The pressure increased, and as the mercury ascended in k the water rose in the tube cdg. By means of the cocks e and D, the escape of the air and water was so regulated that the water height in cdg remained constant. When this height was about three feet over the surface of water in the bottle, the opening being 0-4 of an inch, a multitude of small bubbles were seen to descend with the water through the entire breadth of the tube. When the diameter was greater, the motion was quicker, so that the course of the bubbles could not be exactly followed. When, on the contrary, the opening b was smaller, e.g., 0-2 of an inch in diameter, the air bubbles were observed at f, but they could not penetrate to the lower end of the tube; as soon as they had attained a certain depth, they rose again in consequence of their small specific gravity. Small isolated bubbles only were able to penetrate to a depth of 24 inches.

545. It is evident that the bubbles are formed at the point where the falling water meets the surface f; here they are quite inclosed by the water, and carried with it downwards. If the force which causes this motion be so great that the bubbles descend more quickly than they would arise in consequence of their small specific gravity; then they will reach the bottle AB. This, however, can only take place when the water falls from a sufficient height, and when the opening b bears a sensible ratio to the diameter of the tube cdg. If this ratio be small, the motion of the water in the latter is feeble, and the bubbles move more quickly upwards than downwards. If the opening be not much smaller than the tube cdg, the falling water closes up the latter, even when the tube, instead of reaching to g, ends immediately under the cork d. The water then stands at a height within the tube corresponding to the air-pressure in the bottle underneath; and, in general, exactly the same effects are observed as when the tube cd penetrates the surface of the water. Hence, in the case of the water-bellows it is unnecessary that the tubes A, A (fig. 110), should reach under the surface of water in the drum CC.

546. It will be observed that the action which takes place here is similar to that which is observed when water is poured into a glass, in which case also bubbles are carried downwards. The phenomena are so far explained by Prof. Magnus in his experiments on the motion of fluids (416).

547. We have an example of a trompe on a large scale in the method sometimes pursued after an explosion in a mine. The nearest rivulet being allowed to flow down the shaft, a considerable volume of fresh air enters with the water, and renders practicable a descent into the mine to succour the injured miners. (See Commerce of Arts; Wollf Opera Mathematica; Richesse Minérale, tom. iii.; Manuel de la Métallurgie du Fer, by M. Karsten, tom. ii.; Annales des Mines, tom. ix., 1824; xi., 1825; Nicholson's Journal, vols. ii. and xii.)

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1 Exhibition of Arts and Manufactures, vol. ii., p. 97. 2 Gilbert's Annalen, vol. iii., p. 125.