1. HYDRODYNAMICS, from ῥῶσις, "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 17th and 18th centuries.

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 De Ljuidentibus Humido, about 250 years before the birth of Christ, and were afterwards applied to experiments by Marinus Ghetaldus in his Archimedes Promotus. 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 compression, the syphon, and the forcing pump, were invented by Ctesibius and Hero; and though these machines 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 wheel or Noria, which was common at that time, and which was a kind of chain pump, consisting of a number of earthen 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 Aquaeductibus urbis Romae Commentarius contains all the hydraulic knowledge of the ancients. After describing the Roman aqueducts, 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 adutages, 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.

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, in which he gave a very satisfactory explanation of several phenomena in the motion of fluids. 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 adutage, 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, in his treatise De Motu Gravium naturaliter accelerato. It was afterwards confirmed by the experiments of Raphael Magiotti, on the expence of water discharged from different adutages 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 preasure of the atmosphere, a treatise on the equilibrium of fluids was found among his manuscripts, and was given to the public in 1662. In the hands of Pascal, hydrodynamics 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 cotemporary 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 Apennines 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 hence a greater number of works were written on the subject in Italy than in all the rest of Europe.

11. Guglielmini was the first who attended to the Theory of motion of water in rivers and open canals. Embarrassed 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 contradictory 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 investigation he has shewn 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 being a hyperboloid 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 irreconcileable 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 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 2 to 1. 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 effluent water as due to the whole height of water in the reservoir; and by this means his theory became more conformable to the results of experience. This theory however is still liable to serious objections. The formation of a cataract is by no means agreeable to the laws of hydrostatics; for when a vessel is emptied by the efflux of water through an orifice in its bottom, all the particles of the fluid direct themselves toward this orifice, and therefore no part of it can be considered as in a state of repose.

13. The subject of the oscillation of waves, one of the most difficult in the science of hydrodynamics, was first investigated by Sir Isaac Newton. By the 44th proposition of the 2d book of his Principia, he has furnished us with a method of ascertaining the velocity of the waves of the sea, by observing the time in which they rise and fall. If the two vertical branches of a syphon which communicate by means of a horizontal branch be filled with a fluid of known density, the two fluid columns when in a state of rest will be in equilibrium and their surfaces horizontal. But if the one column is raised above the level of the other, and left to itself, it will descend below that level, and raise the other column above it; and after a few oscillations, they will return to a state of repose. Newton occupied himself in determining the duration of these oscillations, or the length of a pendulum isochronous to their duration; and he found by a simple process of reasoning, that, abstracting from the effects of friction, the length of a synchronous pendulum is equal to one-half of the length of the syphon, that is, of the two vertical branches and the horizontal one, and hence he deduced the isochronism of these oscillations. From this Newton concluded, that the velocity of waves formed on the surface of water either by the wind or by means of a stone, was in the subduplicate ratio of their size. When their velocity therefore is measured, which can be easily done, the size of the waves will be determined by taking a pendulum which oscillates in the time that a wave takes to rise and fall.

14. In the year 1718 the Marquis Poleni published at Padua his work De Castellis per quae derivantur the Marques Poleni's Laws of Fluids, &c. He found from a great number of experiments, that if A be the aperture of the orifice, and D its depth below the surface of the reservoir, the quantity of water discharged in a given time will be as \( 2 \text{AD} \times \frac{0.571}{1,000} \), while it ought to be as \( 2 \text{AD} \), if 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 considered 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 Bernouilli published his Hydrodynamica, seu de viribus et motibus Fluidorum Commentarii. His theory of the motion of fluids was founded on two suppositions, which appeared to him conformable to experience. He supposed 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 virtutum virium, and obtained very elegant solutions. In the opinion of the abbé Boffin, his work was one of the finest productions of mathematical genius.

16. The uncertainty of the principle employed by Daniel Bernouilli, which has never been demonstrated by Maclaurin in a general manner, deprived his results of that confidence which they would otherwise have deserved; and John Bernouilli, who resolved the problem by more direct methods, the one in his Fluxions, published in 1742; and the other in his Hydraulica 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 Bernouilli is, in the opinion of La Grange, defective in perspicuity and precision.

17. The theory of Daniel Bernouilli was opposed by the celebrated D'Alembert. When generalizing James Bernouilli's Theory of Pendulums, he discovered a principle of dynamics so simple and general, that it reduced the laws of the motion 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 late 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 defedraturum 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 réflexion des fluides. It was brought to perfection in his Oeuvres Mathematiques, 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 formulae restrained 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 five to twenty-two feet, from a tower constructed of the finest masonry. Basins 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 are, in every respect, entitled to our confidence.

19. The experiments of the Abbé Boffut, whose labours in this department of science have been very arduous 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. Boffut employed a glass cylinder, to the bottom of which different apertures 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 bales in the ratio of 3 to 2.—It appears also, from the experiments of Boffut, that when water issues through an orifice made in a thin plate, the expence of water, as deduced from theory, is to the real expence 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 expence 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 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 Boffut, 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 Boffut was directed to a variety of other interesting points, which we cannot stop to notice, but for which, must refer the reader to the works of that ingenious author.

20. The oscillation of waves, which was first discussed by Sir Isaac Newton, and afterwards by D'A. M. Flau- lemberg, in the article Ondes, in the French Encyclopaedia, was now revived by M. Flaugergues, who attempted to overthrow the opinions of these philosophers. He maintained, that a wave is not the effect of a motion in the particles of water, 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.

21. This difficult subject has also been discussed by And of M. de la Grange, in his Mecanique Analytique. He de la found, that the velocity of waves, in a canal, is equal Grange, to that which a heavy body would acquire by falling through a height equal to half the depth of the water Part I.

Hydrostatics in the canal. If this depth, therefore, be one foot, the velocity of the waves will be 5.493 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 La Grange 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.

22. The most successful labourer in the science of hydrodynamics, was the chevalier Buat, 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 his Principes d'Hydraulique, which contains a satisfactory theory of the motion of fluids founded solely upon experiments. The chevalier du Buat 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. Buat, therefore, affirms 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. Buat, 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. Buat 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. All these experiments he arranged under some circumstances of resemblance, and produced the following proposition, which agrees in a most wonderful manner with the immense number of facts which he has brought together, viz. \( V = \frac{307 \times \sqrt{d-0.1}}{\sqrt{s-1} \cdot \sqrt{s+1.6}} - 0.3 \times \frac{d}{d-0.1} \), where \( d \) is the hydraulic mean depth, \( s \) the slope of the pipe, or of the surface of the current, and \( V \) the velocity with which the water issues. The theory of M. Buat, with its application to practice, will be found in the articles RIVER and WATER-Works.

23. M. Venturi, professor of natural philosophy in the university of Modena, has lately brought to light some curious facts respecting the motion of water, in his work turi. 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 explained many important facts in hydraulics. He has not attempted to explain this principle; but has shewn, that the mutual action of the fluid particles does not afford a satisfactory explanation of it. The work of Venturi contains many other interesting disquisitions, which are worthy of the attention of every reader.

24. The science of hydrodynamics has of late years been cultivated by M. Eytelwein of Berlin, whose practical conclusions coincide nearly with those of Bossut;—Eytelwein by Dr Matthew Young, late bishop of Clonfert, who has explained the cause of the increased velocity of efflux through additional tubes, and by Mr Vince, Dr T. Young, Coulomb, and Don George Juan; but the limits of this work will not permit us to give any further account of their labours at present. We must now proceed to initiate the reader into the science itself, beginning with that branch of it which relates to the pressure, equilibrium, and cohesion of non-elastic fluids.

PART I. HYDROSTATICS.

Definition of hydrostatics.

25. 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 (A), and the phenomena of capillary attraction.

Definitions and Preliminary Observations.

26. 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.

27. 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

(a) The discussion of this subject is reserved as an introduction to the article SHIP-Building. Hydrodynamics.

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 conglobulate 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.

28. It was universally believed, till within the last 45 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 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 (b). This 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.

29. About the year 1761, the ingenious Mr Canton began to consider this subject with attention, and diffusing the result obtained by the academy del Cimento, resolved to bring the question to a decisive issue (c). 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 flood 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 flood \( \frac{1}{30} \)th 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 flood \( \frac{4}{30} \)th 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 \( \frac{1}{30} \)th 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.

<table> <tr> <th>Millionth Parts.</th> <th>Specific Gravity.</th> </tr> <tr> <td>Compression of mercury,</td> <td>3</td> <td>13.595</td> </tr> <tr> <td>sea-water,</td> <td>40</td> <td>1.028</td> </tr> <tr> <td>rain-water,</td> <td>46</td> <td>1.000</td> </tr> <tr> <td>oil of olives,</td> <td>48</td> <td>0.918</td> </tr> <tr> <td>spirit of wine,</td> <td>66</td> <td>0.846</td> </tr> </table>

Left 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.

30. The experiments of Mr Canton have been lately confirmed by Professor Zimmerman. He found that sea-water was compressed \( \frac{1}{30} \)th 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 high. 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 hence their division into elastic and non-elastic is equally improper.

31. 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)." D'Alembert at first (e) 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

(b) Bacon's works, by Shaw, vol. ii. p. 521. Novum Organum, part ii. sect. 2. aph. 45. § 222. (c) See the Philosophical Transactions for 1762 and 1764, vols iii. and iv. (d) Nov. Comment. Petropol. tom. xiii. p. 305. (e) Mélanges de Litterature, d'Histoire, et Philosophie. afterwards* adopted the same property as Euler from the foundation which it furnishes for an algebraical calculus. The same property has been employed by Buffon, Prony, and other writers, and will form the first proposition of the following chapter.

CHAP. I. On the Pressure and Equilibrium of Fluids.

PROPOSITION I.

32. When a mass of fluid, supposed without weight, is subjected to any pressure, that pressure is so diffused throughout the whole, that when it remains in equilibrium all its parts are 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 m n, o p 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 m n, o p 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 o p, by means of the piston B, will propagate itself through every part of the circular vessel EF, so that if the orifices m n, t u are shut, and r s open, the fluid would rush through this aperture in the same manner as it would rush through m n or t u, 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 in water, an equilibrium may exist even when a particle is left pressed in one direction than in another; but this inequality of pressure is so exceedingly trifling, that the proposition may be considered as true, even in cases of imperfect fluidity.

PROP. II.

33. If to the equal orifices m n, t u, o p, r s of a vessel, containing a fluid destitute of weight, be applied equal powers A, B, C, D, in a perpendicular direction, or if the orifices m n, &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 m n, r s, t u, that the pressure of the power C is transmitted equally to the orifices m n, o p, t u, and so on with all the 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 m n, o p, r s, t u; but if A : B = m n : o p, and so on with the rest, the fluid will still be in equilibrium. Let A be greater than B, then m n will be greater than o p; and whatever number of times B is contained in A, so many times will o p be contained in m n. If A = 2 B, then m n = 2 o p, and since the orifice m n is double of o p, 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 = 2 B, 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 shewn in the same way that the fluid will be in equilibrium.

PROP. III.

34. The surface of a fluid, influenced by the force of gravity and in equilibrium in any vessel, is horizontal, or at right angles to the direction of gravity.

Let the surface of the fluid be supposed to assume the waving form APEB. Any particle P in the surface of the fluid is influenced by the force of gravity, which may be represented by PS, and which may be decomposed into two forces P m, P n in the direction of the two elementary portions of the surface P m, P n. (See Dynamics, 148.) But since the particle P is in a state of equilibrium, the force of gravity acting in the direction P m, P n must be destroyed by equal and opposite forces, exerted by the neighbouring particles against P in the direction m P, n P; therefore the forces P m, P n are equal to the forces m P, n P. Now the particle P being in equilibrium, must be equally pressed in every direction (32.). Wherefore the forces P m, P n are equal, and by the doctrine of the composition of forces, (see Dynamics, 133. D), the angle m P n formed by the two elementary portions P m, P n of the surface of the fluid, must be bisected by PS, the line which represents the direction of gravity. The same may be proved of every other point of the surface of the fluid; and therefore this surface must be horizontal or perpendicular to the direction of gravity.

35. This proposition may be otherwise demonstrated. From the principles of mechanics, it is obvious, that when the centre of gravity of any body is at rest, the body itself is at rest; and that when this centre is not supported, the body itself will descend, till it is prevented by some obstacle from getting farther. In the same manner the centre of gravity of a fluid mass will descend to the lowest point possible; and it can be shewn that this centre will be in its lowest position when the surface of the fluid mass is horizontal. For let FGHI (fig. 2.) be any surface, whether solid or fluid, and C its centre of gravity, the point C is nearer the line HI when FG is parallel to HI and rectilineal, than when it has any other form or position. When the surface FGHI is suspended by the point C, or balanced upon it, it will be in equilibrium; but if the line F is made to assume any other form as Fr st G, by removing the portion G o p of the surface to r s t, the equilibrium equilibrium will be destroyed, and the fide FG will preponderate. In order, therefore, to restore the equilibrium, the surface must be balanced on a point c farther from HI; that is, the centre of gravity of the surface Frstop IH is c. In the same way it may be shewn, that whatever be the form of the bounding line FG, the quantity of surface remaining the same, its centre of gravity will be nearest HI, when FG is rectilineal and parallel to it.—On the truth contained in this proposition depends the art of levelling, and the construction of the spirit level, for an account of which see Levelling.

36. 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 the account.

Prop. IV.

37. The surface of a fluid influenced by the force of gravity, and contained in any number of communicating vessels, however different in form and position, will be horizontal.

Let ABCDE be a system of communicating vessels into which a quantity of fluid is conveyed: It will rise to the same height in each vessel, and have a horizontal surface ABCDE. Suppose AGFE a large vessel full of water. By the last proposition, its surface ABCDE will be horizontal. Now, if any body be plunged into this vessel, the cylinder C for instance, the surface of the fluid will still be horizontal; for no reason could be assigned for the water's rising on one side of this body any more than on another. Let us now take out the cylinder C, and immerse into the fluid, successively, the solid bodies A a, B b, C c, D d, then after each immersion the surface will still be horizontal; and when all these solids are immersed, the large vessel AF will be converted into the system of communicating vessels represented in fig. 4.; in which the surface of the fluid will, of consequence, be horizontal.

38. This proposition may be also demonstrated by supposing the parts A a, B b, C c, D d, converted into ice without changing their former magnitude. When this happens, the equilibrium will not be disturbed; and the fluid mass AF, whose surface was proved to be horizontal by the last proposition, will continue in the same state after the congelation of some of its parts. That is, the surface of the fluid in the communicating vessels A, B, C, D, E will be horizontal.

39. When the communicating vessels are so small that they may be regarded as capillary tubes, the surface of the fluid will not be horizontal. From the attraction which all fluids have for glass, they rise to a greater height in smaller tubes than in larger ones, and the quantity of elevation is in the inverse ratio of the diameters of the bores. In the case of mercury, and probably of melted metals, the fluid substance is depressed in capillary tubes, and the depression is subject to the same law. The subject of capillary attraction will be treated at length in a subsequent part of this article.

40. 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.

Prop. V.

41. If a mass of fluid contained in a vessel be in equilibrium, any particle whatever is equally prefed in every direction, with a force equal to the weight of a column of particles whose height is equal to the depth of the particle prefed below the surface of the fluid.

Immerse the small glass tube mp, into the vessel AB Fig. 5. filled with any fluid; then if the tube is not of the capillary kind, the fluid will rise to n on the same level with the surface AB of the fluid in the vessel. Now it is evident, that the particle p at the bottom of the tube mp is prefed downwards by the superincumbent column of particles np, which is equal to the depth of the particle p below the surface of the fluid. But since the mass of fluid is in equilibrium, the particle p is prefed equally in every direction: Therefore, the particle p is prefed equally in every direction by a force equal to the superincumbent column np.

Prop. VI.

42. A very small portion of a vessel of any form, filled with a fluid, is prefed with a force which is in the compound ratio of the number of particles contained in that surface, its depth below the surface of the fluid, and the specific gravity of the fluid.

Let Dp EB be the vessel, and rs a very small portion of its surface, the prefure upon rs is in the compound ratio of the number of particles in rs, and np its depth below the horizontal surface DB. Suppose the glass tube mp to be inserted in the infinitely small aperture p, then, abstracting from the influence of capillary attraction, the fluid in the glass tube will ascend to m on a level with DB, the surface of the fluid in the vessel, and the particle p will be prefed with a column of particles, whose height is np. In the same way it may be shewn, that every other particle contained between r and s is prefed with a similar column. Then, since \( p \times n p \) will represent the prefure of the column np on the particle p; if N be the number of particles in the space rs, \( N \times n p \) will be the force of the column supported by the space rs. And as the weight of this column must increase with the specific gravity of the fluid, \( S \times N \times n p \) will represent its prefure, S being the specific gravity of the fluid.

Prop. VII.

43. The prefure upon a given portion of the bottom of a vessel, whether plane or curved, filled with any any fluid, is in the compound ratio of the area of that portion, and the mean altitude of the fluid, that is, the perpendicular distance of the centre of gravity of the given portion from the surface of the fluid; or, in other words, the pressure is equal to the weight of a column of fluid whose base is equal to the area of the given portion, and whose altitude is the mean altitude of the fluid.

Let AEGB be the vessel, and AFB the surface of the fluid which it contains. Let GH be a given portion of its bottom, and C the centre of gravity of that portion: Then shall CF be the mean altitude of the fluid.—Conceive the portion GH to be divided into an infinite number of small elements Hh, Gg, &c. then (42.) the pressure sustained by the elements Hh, Gg, will be respectively S × Hh × Hw; S × Gg × Gt, &c. the specific gravity of the fluid being called S. But it follows from the nature of the centre of gravity, that the sum of all these products is equal to the product of the whole portion GH into CF the distance of its centre of gravity from the horizontal surface of the fluid (f). Therefore the pressure upon the portion GH is in the compound ratio of its surface converted into a plane, and the mean altitude of the fluid.

44. From this proposition we may deduce what is generally called the Hydrostatic paradox, viz. that the pressure upon the bottoms of vessels filled with fluid does not depend upon the quantity of fluid which they contain, but upon its altitude; or, in other words, that any quantity of fluid, however small, may be made to balance any quantity or any weight, however great. Let ACOQRDPB be a vessel filled with water, the bottom QR will sustain the same pressure as if it supported a quantity of water equal to MQRN. It is evident (43.) that the part EF is prefixed with the column of fluid ABEF, and that the part DG equal to CD is pushed upwards with the weight of a column equal to ABCD. Now, as action and reaction are equal and contrary, the part DG reacts upon FH with a force equal to the weight of the column ABCD, and FH evidently sustains the smaller column DGFH; therefore FH sustains a pressure equal to the weight of the two columns ABCD and DGFH, that is, of the column BIHF. In the same way it may be shewn, that any other equal portion of the bottom QR sustains a similar pressure; and therefore it follows, that the pressure upon the bottom QR is as great as if it supported the whole column MNQR.

45. The same truth may be deduced from Prop. IV.

For since the fluid in the two communicating vessels AB, CD will rise to the same level, whatever be their size, the fluid in AB evidently balances the fluid in CD; and any surface mn is prefixed with the same force in the direction Bm by the small column AB, as it is prefixed in the direction Dm by the larger column CD.

46. Cor. 1. From this proposition it follows, that the whole pressure on the fides 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 fides 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 fides and bottom will be equal to three times that weight.

Cor. IV. The pressure sustained by different parts of the fide of a vessel are as the squares of their depths below the surface; and if their depths are made the abscissa of a parabola, its ordinate will indicate the corresponding pressures.

DEFINITION.

47. 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, the force applied and the total pressure would be in equilibrium.

PROP. VIII.

48. The centre of pressure coincides with the centre of percussion.

Let AB be a vessel full of water, and CE the section of a plane whose centre of pressure is required, centre of percussion CE till it cuts the surface of the water in M, pressure. Take any point D, and draw DO, EP, CN perpendicular to the surface MP. Then if M be made the axis of suspension of the plane CE, the centre of percussion

(f) This will be evident from the following proposition. If every indefinitely small part of a surface be multiplied by its perpendicular distance from a given plane, the sum of the products will be equal to the product of the whole surface, multiplied by the perpendicular distance of its centre of gravity from the same plane. In Plate CCLXIII. fig. 7, let a, c represent two weights suspended at their centre of gravity by the lines a A, c C attached to the horizontal plane of which ABC is a section, and let b be the common centre of gravity of these weights and b B the distance of this centre from the given plane, then \( a \times a\ A + c \times c\ C = a + c \times b\ B \).—Draw a n, c m at right angles to b B. Then since b is the common centre of gravity of the weights a, c, we shall have by the similar triangles a n b, c m b (Euclid VI.4.) \( n\ b : m\ b = (b\ a : b\ c) : c : a \) (See MECHANICS, Centre of Gravity). Hence \( a \times n\ b - c \times m\ b \), or \( a \times n\ B - b\ B - c \times b\ B - m\ B \), or \( a \times n\ b - a \times b\ B - c \times b\ B - c \times m\ B \); then, by transposition, \( a \times n\ b - c \times m\ B = a \times b\ B + c \times b\ B - a + c + b\ B \). But \( n\ B = a\ A \) and \( m\ B = c\ C \), therefore, by substitution \( a \times a\ A + c \times c\ C = a + c \times b\ B \). By supposing the two weights a and c united in their common centre of gravity, the same demonstration may be extended to any number of weights. of the plane CE revolving round M will also be the centre of preasure. If MCE moves round M as a centre, and strikes any object, the percussive force of any point C is as its velocity, that is, as its distance CM from the centre of motion; therefore the percussive force of the points C, D, E, are as the lines CM, DM, EM. But the preasures upon the point C, D, E, are as the lines CN, DO, EP, and these lines are to one another as CM, DM, EM; therefore the percussive forces of the points C, D, E, are as the preasures upon these points. Consequently, the centre of preasure will always coincide with the centre of percussion.

Sect. II. Instruments and Experiments for illustrating the Preasure of Fluids.

49. We have already shewn in art. 41. that the preasure upon the bottoms of vessels filled with fluids does not depend upon the quantity of fluid which they contain, but upon its particular altitude. This proposition has been called the Hydrostatical Paradox, and is excellently illustrated by the following machine. In fig. 1. AB is a box which contains about a pound of water, and a b c d a glass tube fixed to the end C of the beam of the balance, and the other end to a moveable bottom which supports the water in the box, the bottom and wire being of an equal weight with an empty scale hanging at the other end of the balance. If one pound weight be put into the empty scale, it will make the bottom rise a little, and the water will appear at the bottom of the tube a, consequently it will prees with a force of one pound upon the bottom. If another pound be put into the scale, the water will rise to b, twice as high as the point a, above the bottom of the vessel. If a third, a fourth, and a fifth pound be put successively into the scale, the water will rise at each time to c, d, and e, the divisions ab, bc, cd, de being all equal. This will be the case, however small the bore of the glass tube; and since when the water is at b, c, d, e, the preasures upon the bottom are successively twice, thrice, four times, and five times as great as when the water was contained within the box, we are entitled to conclude that the preasure upon the bottom of the vessel depends altogether on the altitude of the water in the glass tube, and not upon the quantity it contains. If a long narrow tube full of water, therefore, be fixed in the top of a cask likewise full of water, then though the tube be so small as not to hold a pound of the fluid, the preasure of the water in the tube will be so great on the bottom of the cask as to be in danger of bursting it; for the preasure is the same as if the cask was continued up in its full size to the height of the tube, and filled with water. Upon this principle it has been affirmed that a certain quantity of water, however small, may be rendered capable of exerting a force equal to any assignable one, by increasing the height of the column, and diminishing the base on which it prees. This, however, has its limits; for when the tube becomes so small as to belong to the capillary kind, the attraction of the glass will support a considerable quantity of the water it contains, and therefore diminish the preasure upon its base.

50. The preceding machine must be so constructed, that the moveable bottom may have no friction against the inside of the box, and that no water may get between it and the box. The method of effecting this will be manifest from fig. 2. where ABCD is a section of the box, and a b c d its lid, which is made very light. The moveable bottom E, with a groove round its edges, is put into a bladder f g, which is tied close around it in the groove, by a strong waxed thread. The upper part of the bladder is put over the top of the box at a and d all around, and is kept firm by the lid a b c d, so that if water be poured into the box through the aperture l l in its lid, it will be contained in the space f E g h, and the bottom may be raised by pulling the wire i fixed to it at E.

51. The upward preasure of fluids is excellently illustrated by the hydrostatic bellows. The form given to this machine by the ingenious Mr Ferguson (Lecturer, vol. ii. p. 111.) is represented in fig. 3. where ABCD is an oblong square box, into one of whose hydrofides is fixed the upright glass tube a I, which is bent into a right angle at the lower end as at i, fig. 4. To this bent extremity is tied the neck of a large bladder K, which lies in the bottom of the box. Over this bladder is placed the moveable board L, figs. 3. and 4. in which the upright wire M is fixed. Leaden weights NN, with holes in their centre, to the amount of 16 pounds, are put upon this wire, and prees with all their weight upon the board L. The cross bar p is then put on, in order to keep the glass tube in an upright position; and afterwards the piece EFG for keeping the weights NN horizontal, and the wire M vertical. Four upright pins, about an inch long, are placed in the corners of the box, for the purpose of supporting the board L, and preventing it from preesing together the sides of the bladder. When the machine is thus fitted up, pour water into the tube I till the bladder is filled up to the board L. Continue pouring in more water, and the upward preasure which it will excite in the bladder will raise the board with all the weights NN, even though the base of the tube should be so small as to contain no more than an ounce of water.

52. That the preasure of fluids arises from their gravity, and is propagated in every direction, may be proved by the following experiment. Imbibe into an empty vessel, a number of glass tubes bent into various angles. Into their lower orifices introduce a quantity of mercury, from which will rest in the longer legs on a level with the gravity, and orifices. Let the vessel be afterwards filled with water, and it will be seen, while the vessel is filling, that the mercury is gradually preesed from the lower orifices towards the higher, where the water is prevented from entering. Now, in consequence of the various angles into which the glass tubes are bent, the lower orifices point to almost every direction; and therefore it follows, that the preasure of the superincumbent water is propagated in every direction. When a straight tube is employed to shew the upward preasure of fluids, the mercury which is introduced into its lower extremity must be kept in by the finger till the height of the water above the orifice is equal to fourteen times the length of the column of quicksilver: When the finger is removed the mercury will ascend in the tube.

53. The preasure of the superior strata of fluids upon the inferior strata may be shewn in the following manner. Immerse two tubes of different bores, but not of prees upon the capillary kind, in a vessel of mercury. The mercury will rise in the tube on a level with its surface in the vessel. Let water then be poured upon the mercury so as not to enter the upper orifices of the tubes, the pressure of the water upon the inferior fluid will cause the mercury to ascend in the tubes above the level of that in the vessel, but to the same height in both tubes. The columns of quicksilver in the two tubes are evidently supported by the pressure of the water on the inferior fluid. The same experiment may be made with oil and tinged water, the latter being made the inferior fluid.

54. The syphon is an instrument which shews the gravitation of fluids, and is frequently employed for decanting liquors. It is nothing more than a bent tube EABC F, having one of its legs longer than the other. The shorter leg BCF is immersed in the fluid contained in the vessel D; and if, by applying the mouth to the orifice E, the air be sucked out of the tube, the water in the vessel D will flow off till it be completely emptied. Now it is obvious that the atmosphere which has a tendency to raise the water in the shorter leg EB by its pressure on the surface of the water at C, has the same tendency to prevent the water from falling from the orifice E, by its pressure there, and therefore if the syphon had equal legs as AB, BC, no water could possibly issue from the orifice E. But when the leg EB is longer than BC, the column of fluid which it contains being likewise longer, will by its superior weight cause the water to flow from the orifice E, and the velocity of the issuing fluid will increase as the difference between the two legs of the syphon is made greater.

55. In order to shew that the effect of the syphon depends upon the gravitations of fluids, M. Paical devised the following experiment. In the large glass vessel AB, fasten by means of bees wax two cylindrical cups a, b, containing tinged water, whose surface is about an inch higher in the one than in the other. Into the tinged water insert the legs of a glass syphon c d, having an open tube e fixed into the middle of it, and put a wooden cover on the vessel with a hole in its centre to receive the tube and keep it in a vertical position. Then through the funnel f, fixed in another part of the cover, pour oil of turpentine into the larger vessel till it flow into the cups a, b, and rise above the arch of the syphon. The pressure of the oil upon the tinged water in the cups will cause the water to pass through the syphon from the higher cup to the lower, till the surfaces of the water in both the cups be reduced to a level. In order to explain this, suppose a horizontal plane e b to pass through the legs of the syphon, and the tinged water in the cups, the parts of this plane within the legs when the syphon is full, will be equally pressed by the columns of tinged water c e, d b within the syphon; but the equal parts of this plane between the circumference of each leg of the syphon, and the circumference of each cylindrical cup, their diameters being equal, will sustain unequal pressures from their superincumbent columns, though the altitudes of these columns be equal. For since the pressure upon e is exerted by a column of oil a c, and a column of water a e, whereas the pressure upon b is exerted by a column of oil h d, and a column of water h b; the column c e which contains the greatest quantity of water, will evidently exert the greatest force, and by its pressure will drive the tinged water from the cup a, through the syphon a c d into the cup b, until a perfect equilibrium is obtained by an equality between the columns of water a e and h b.

SECT. III. Application of the Principles of Hydrostatics to the Construction of Dykes, &c. for resisting the pressure of water.

DEFINITION.

A dyke is an obstacle either natural or artificial, which opposes itself to the constant effort of water to spread itself in every direction.

56. In discussing this important branch of hydraulic architecture, we must inquire into the thickness and form which must be given to the dyke in order to resist the pressure of the water. In this inquiry the dyke may be considered as a solid body which the water pressure of tends to overthrow by turning it round upon its posterior angle C; or it may be regarded as a solid, whose foundation is immovable, but which does not resist the pressure of the water through the whole of its height, and which may be separated into horizontal sections by the efforts of the fluid. A dyke may be considered also as a solid body which can be neither broken nor overturned, but which may be pushed horizontally from its base, and can preserve its stability only by the friction of its base on the ground which supports it. On these conditions are founded the calculations in the following proposition which contain the most useful information that theory can suggest upon the construction of dykes.

PROP. I.

57. To find the dimensions of a dyke which the water tends to overthrow by turning it round its posterior angle.

Let ABCD be the section of the dyke, considered as a continuous solid, or a piece of firm masonry, HK the level of the water which tends to overthrow it by turning it round its posterior angle C, supposed to be fixed, and let AC, BD, be right lines or known curves. It is required to determine CD the thickness which must be given to its base to prevent it from being overturned.

To the surface of the water HK draw the ordinates PM, p m infinitely near each other, and let fall from the points H and M the perpendiculars HT, MX. Draw the horizontal line ML and raise the perpendicular CL, and suppose

<table> <tr><th>HP</th><th>= x</th></tr> <tr><th>PM</th><th>= y</th></tr> <tr><th>Pp or MV the fluxion of x</th><th>= z</th></tr> <tr><th>Vm the fluxion of y</th><th>= s</th></tr> <tr><th>HT</th><th>= a</th></tr> <tr><th>DT</th><th>= b</th></tr> <tr><th>CD</th><th>= c</th></tr> <tr><th>The momentum of the area ABCD, or the force with which it resists being turned round the fulcrum C</th><th>= Z</th></tr> <tr><th>The specific gravity of water</th><th>= s</th></tr> <tr><th>The specific gravity of the dyke</th><th>= r</th></tr> </table>

58. It is obvious from art. 41, that every element sustains a perpendicular pressure proportional to the height PM. Let RM perpendicular to MM represent the force exerted by the column of water \( M m p P \), and let it be decomposed into two other forces, one of which \( R Q \) is horizontal, and has a tendency to turn the dyke round the point \( C \), and the other \( R Y \) is vertical and tends to press the dyke upon its base. The force \( R Q \) is evidently \( s y \times M m \times \frac{R Q}{RM} \), (42) and therefore the horizontal part of it will be only \( s y \times M m \times \frac{R Q}{RM} \). But the triangles \( R Q M, M V m \) are evidently similar, consequently \( R Q : RM = V m : M m \); hence \( \frac{R Q}{RM} = \frac{V m}{M m} = \frac{\dot{y}}{\dot{m}} \). Wherefore by substitution we have the force \( R Q = s y \times M m \times \frac{\dot{y}}{\dot{M} m} \), and dividing by \( M m \), we have \( R Q = s y \dot{y} \). The force \( R Q \), therefore, will always be the same as the force against \( V m \), whatever be the nature of the curve BD. Now the momentum of this force with relation to the fulcrum \( C \), or its power to make the dyke revolve round \( C \), is measured by the perpendicular CL let fall from the centre of motion to the direction in which the force is exerted (See MECHANICS), consequently this momentum will be \( s y \dot{y} \times CL = s y \dot{y} \times a - y \) (since \( CL = HT - PM = a - y \))\( = s a y \dot{y} - s y \dot{y} \), whose fluent is \( \frac{s a y \dot{y}}{2} - \frac{s y \dot{y}^2}{3} \), which by supposing \( y = a \) becomes \( \frac{1}{6} s a^3 \) for the total momentum of the horizontal effort of the water to turn the dyke round \( C \). The vertical force \( R Y \) or \( Q M \), which presses the dyke upon its base, is evidently \( s y \times M m \times \frac{MQ}{RM} \), but on account of the similar triangles \( \frac{MQ}{RM} = \frac{\dot{x}}{\dot{M} m} \); consequently by substitution we shall have the force \( R Y = s y \times M m \times \frac{\dot{x}}{\dot{M} m} = s y \dot{x} \), after division by \( M m \). The momentum, therefore, of the vertical force \( R Y \) with relation to \( C \), or its power to prevent the dyke from moving round the fulcrum \( C \), will be \( s y \dot{x} \times CX \); \( CX \) being the arm of the lever by which it acts, or the perpendicular let fall from the fulcrum upon the direction of the force. Now \( CX = CD - DT + TX \) or HP, that is \( CX = x - b + x \); therefore the momentum of the force \( R Y = s y \times x - b + x \), and the sum of the similar moments from \( F \) to \( H \) will be the fluent \( \int (x - b + x) s y \dot{x} \), the combined momentum of all the vertical forces which resist the efforts of the horizontal forces to turn the dyke round \( C \). But the efforts of the horizontal forces are also resisted by the weight of the dyke whose momentum we have called \( Z \), therefore \( \sigma Z \), \( \sigma \) being the specific gravity of the dyke, will be the momentum of the dyke. We have now three forces acting at once, viz. the horizontal force of the water striving to overturn the dyke, and the vertical force of the water combined with the momentum of the dyke, striving to resist its overthrow, therefore we shall have an equilibrium between these three forces, when the momentum of the horizontal forces is made equal to the momentum of the vertical forces, added to that of the dyke itself, consequently

\[ \frac{1}{6} s a^3 = \int (x - b + x) s y \dot{x} + \sigma Z. \]

59. As it is necessary, however, to give more stability to the dyke than what is just requisite to preserve its equilibrium, we must make its dimensions such as to resist a force greater than the horizontal forces, a force, for example, \( n \) times the momentum of the horizontal forces (\( c \)). The equation will therefore become

(I.) \( n \times \frac{1}{6} s a^3 = \int (x - b + x) s y \dot{x} + \sigma Z \),

which comprehends every possible case of stability; for if we wish the stability of the dyke to have double the stability of equilibrium, we have only to make \( n = 2 \). The preceding general equation is susceptible of a variety of applications according to the nature of the curves which form the sides of the dyke. It is at present worthy of remark that since the momentum of the horizontal forces is always the same whatever be the curvature of the sides AC, BD, and since the momentum of the vertical forces increases as the angle CDH diminishes, it follows that it will always be advantageous to diminish the angle CDH and give as much slope as possible to the sides of the dyke.

60. Let us now consider the conditions that may be necessary to prevent the dyke ABCD from sliding on containing its base CD. Since the base of the dyke is supposed the horizontal, the force which the dyke opposes to the efforts of horizontal efforts of the water arises solely from the frictional adhesion of the dyke to its base, and from the resistance of friction. These two forces, therefore, combined that with the weight of the dyke, form the force which resists the horizontal efforts of the water; an equilibrium will consequently obtain when the three first forces are made equal to the last. But the force of adhesion, and the resistance of friction, being unknown, may be made equal to the weight of the dyke multiplied by the constant quantity \( m \), which must be determined by experience. Now calling A the area of the section ABCD, we shall have \( \sigma A \) for its weight, and \( m \sigma A \) for the resistance which is opposed to the horizontal efforts of the water. But we have already seen that the horizontal forces of the water upon \( M \) are equal to \( s y \dot{y} \), whose fluent \( \frac{1}{6} s a^3 \) (when \( a = y \)) is the sum of all the horizontal forces, consequently when an equilibrium takes place between these opposing forces we shall have

(II.) \( m \sigma A = \frac{1}{6} s a^3 \), or \( A = \frac{\sigma}{m} \times \frac{a^3}{2 m} \).

We might have added to the weight of the dyke the vertical pressure of the water, but it has been neglected for the purpose of having the dyke sufficiently strong to resist an additional force.

61. We

(c) The dimensions of the dyke would be sufficiently strong to resist any additional force by neglecting the term \( \sigma Z \), which represents the vertical pressure of the water tending to keep the dyke upon its base. 61. We shall now proceed to inquire into the form which the general equation assumes when the sides of the dyke are rectilineal. Let AC, BD, fig. 9. be two lines inclined to the horizon under given angles ACD, BDC, and let AB, CD be two horizontal lines. Retaining the construction and symbols in art. 57, let fall AQ, BZ perpendicular to CD, and make AQ = BZ = d, CQ = r, and DZ = r'.

On account of the similar triangles HPM, FTH we shall have \( a : b = y : x \), and therefore \( x = \frac{b y}{a} \). Substituting this value of x, instead of x in the general equation, art. 54. we have \( \int (z - b + x) s y \cdot s y' = \frac{s b^2}{a} (z - b) + \frac{b y}{a} y y' = \frac{s b z y}{2 a} - \frac{s b b y y'}{2 a} + \frac{s b b y^2}{3 a^2} \) (making \( y = a \)) \( \frac{s b z a}{2} - \frac{s b^2 a}{6} \); now the momentum of the dyke ABCD with relation to C, is equal to the whole area of the dyke ABCD collected in its centre of gravity, and placed at the end of a lever whose length is the horizontal distance of that centre of gravity from the fulcrum C. But the area of ABQZ = QZ × ZB = \( \frac{z - r - r}{2} \times d \); the area of the triangle ACQ = \( \frac{CQ \times QA}{2} = \frac{dr}{2} \), and the area of the triangle BZD = \( \frac{DZ \times ZB}{2} = \frac{dr'}{2} \). Now the lever by which the area ABQZ collected in its centre of gravity F, acts upon the fulcrum, is evidently \( CF = CQ + QF = CQ + \frac{1}{2} QZ = r + \frac{z - r - r}{2} \), consequently the momentum by which the area ABCD resists the horizontal forces that conspire to give it a motion of rotation about C will be \( \frac{z - r - r}{2} \times d \times r + \frac{z - r - r}{2} \).

The lever by which the triangle BZD acts, when collected in its centre of gravity I, is evidently Ci; but by the property of the centre of gravity Di = \( \frac{2r'}{3} \), hence Ci = CD — Di = \( \frac{2r'}{3} \), consequently the energy of the triangle BZD to resist the efforts of the water acting horizontally will be \( \frac{dr'}{2} \times \frac{2r'}{3} \). The lever of the triangle ACQ is plainly C = \( \frac{1}{3} CQ = \frac{2r}{3} \), consequently the momentum of ACQ collected in its centre of gravity S will be \( \frac{dr}{2} \times \frac{2r}{3} \). Having thus found the momentum of the rectangle ABQZ, and of the triangles BZD, ACQ, the sum of these moments will be the momentum Z, with which the dyke opposes the horizontal efforts of the water, therefore we shall have

\[ Z = \frac{z - r - r}{2} \times d \times r + \frac{z - r - r}{2} + \frac{dr'}{2} \times \frac{2r'}{3} + \frac{dr}{2} \times \frac{2r}{3} \]

and by multiplication

\[ Z = \frac{d z x}{2} - \frac{d r' x}{2} + \frac{d r' r'}{6} - \frac{d r r}{6}. \]

By substituting this value of Z in the general equation in art. 54. we shall have

\[ \text{(III.) } n \times \frac{1}{6} s a^2 = \frac{s b z a}{2} - \frac{s b b a}{6} + \frac{s d z x}{2} - \frac{\sigma d r' \alpha}{2} + \frac{\sigma d r' r'}{6} - \frac{\sigma d r r}{6}, \]

a quadratic equation which will determine in general both its base z of a dyke when its sides are rectilineal and the angle inclined at any angle to the horizon.

62. When the angle ACQ is a right angle, or when the posterior side AC of the dyke is perpendicular to the horizon, the quantity r becomes = 0, and the last term of the preceding equation in which r appears will vanish, consequently the equation will now be-

\[ \text{(IV.) } n \times \frac{1}{6} s a^2 = \frac{s b z a}{2} - \frac{s b b a}{6} + \frac{s d z x}{2} - \frac{\sigma d r' \alpha}{2} + \frac{\sigma d r' r'}{6}, \]

63. When the angles ACQ and BDZ are both right, the dyke becomes rectangular, with its sides perpendicular to its base. In this case both r and r' become = 0, and therefore all the terms in which they are found will vanish. In this case too DT = b becomes vertical, and therefore the terms in which it appears will likewise vanish. The general equation will now become

\[ \text{(V.) } n \times \frac{1}{6} s a^2 = \frac{\sigma d z x}{2} \text{ a pure quadratic.} \]

64. In order to shew the application of the preceding formulae, and at the same time the advantages of inclining the sides of the dyke, let us suppose the depth of the water and also the height of the dyke to be 18 feet, so that B will coincide with H. Let us also suppose, what is generally the case in practice, that the declivity of the sides is \( \frac{1}{5} \) of their altitude, that is DZ = \( \frac{1}{5} \) BZ. Let the specific gravity of the dyke be to that of water as 12 to 7; and suppose it is wished to make the stability of the dyke twice as great as the stability of equilibrium, that is, to make it capable of resisting a force twice as great as that which it really sustains. Then, upon these conditions, we shall have BZ = HT or \( a = d = 18 \) feet; CQ = DZ = DT, or \( r = r' = b = 3 \) feet; \( s = 7 \); \( \sigma = 12 \), and \( n = 2 \). By substituting these numerical values in the general equation No III. it becomes

\[ z = \frac{45}{36} z = \frac{4599}{39} \text{ feet;} \]

a quadratic equation which after reduction will give \( z = 12 \) feet nearly. When \( z = 12 \), the area of the dyke ABCD will be 162 square feet.

65. Let us now suppose the sides of the dyke to be vertical, the equation No V. will give us \( z = 11 \) feet of inclining 2 inches, which makes the area of the dyke more than 201 square feet. The area of the dyke with inclined sides fides is therefore to its area with vertical fides nearly as 4 to 5: and hence we may conclude that a dyke with inclined fides has the same stability as a dyke with vertical fides; while it requires \( \frac{1}{2} \) less materials.

PROP. II.

To find the dimensions of the dyke when the water tends to separate it into horizontal sections or laminae.

66. To find the dimensions of a dyke which can neither slide upon its base, nor turn round its posterior angle; but which is composed of horizontal sections, which may be separated from each other.

In solving this proposition we must find the curvature of the side exposed to the pressure of the water, which will make all the different sections or horizontal laminae equally capable of resisting the different forces which tend to separate them. If the lamina NM does not resist the column PM, which partly presses it in the direction MN as powerfully as the lamina nm resists the horizontal pressure of the column pm, the lamina NM is in danger of being separated from the lamina nm. But if all the laminae NM, nm resist with equal force the horizontal effects of the water, and if the dyke cannot be made to slide upon its base nor turn round its posterior angle T, it cannot possibly yield to the pressure of the water; for it is impossible to separate one lamina from another, unless the one opposes a less resistance than the other. To simplify the investigation as much as possible, let us suppose the posterior fide of the dyke to be vertical, and the depth of the water to be equal to the height of the dyke.

67. Let ABC be the section of the dyke, AK the surface of the water, AC the curvature required, AB its posterior fide; MN nm a horizontal lamina infinitely small, in the direction of which the dyke has a tendency to break in consequence of the efforts of the water upon AM.

If the dyke should break in the direction MN, the superior part AMN will detach itself from the inferior part MNBC, by moving from M towards N; and at the moment when the impulse takes place it will have a small motion of rotation round the point N. We must therefore determine the forces which act upon the lamina MN nm, and form an equation expressing their equilibrium round the point N. The forces alluded to are evidently, 1. The horizontal efforts of the water; 2. The vertical efforts of the water; 3. The weight of the part AMN; and, 4. The adhesion of the two surfaces MN, mn. Of these four forces the first is the only one which has a tendency to overthrow the portion AMN of the dyke; and its efforts are resisted by the three other forces. In order to find the momenta of these forces with regard to the point N let us suppose

AP=NM \( =x \) PM \( =y \) The specific gravity of water \( =s \) The specific gravity of the dyke \( =\sigma \)

Then we shall have, 1. The momentum of the horizontal forces of the water will be \( \frac{1}{2} sy^2 \), by the same reasoning that was employed in art. 57.

2. The momentum of the part AMN of the dyke will be \( \sigma \int x' y' \) the area of the surface AMN, multiplied by the distance of its centre of gravity from the fulcrum N, which is equal to \( \frac{\int x' x'' y'}{\int x' y'} \). See MECHANICS.

68. In order to simplify the calculus, and at the same time increase the stability of the dyke, we shall neglect the vertical force of the water, and the adhesion of the two surfaces MN, mn. The only forces therefore which we have to consider, are the horizontal efforts of the water acting against the momentum of the superior part AMN. By making an equilibrium between these forces we shall have the following equation

\[ \frac{1}{2} sy^2 = \sigma \int x' y' \times \frac{\int x' x'' y'}{\int x' y'} = \frac{1}{2} \sigma \times \int x' x'' y'. \]

By taking the fluxion we have

\[ \frac{1}{2} sy^2 \dot{y} = \frac{1}{2} \sigma \times x' x'' \dot{y}. \] Dividing by \( \dot{y} \) we have \[ \frac{1}{2} s y^2 = \frac{1}{2} \sigma \times x'^2, \] which by reduction becomes \[ y = \sqrt{\frac{\sigma}{s}} \times x. \]

The line AMC therefore is rectilineal, and the base BC is to the altitude BA as \( \sqrt{s} : \sqrt{\sigma} \); that is, as the square root of the specific gravity of the water is to the square root of the specific gravity of the dyke.

69. In order to prevent the superior portion AMN Equation from sliding on its base MN, we must procure an equilibrium between the adhesion of the surfaces MN, mn, the conditions and the horizontal force exerted by the water. Now equilibrium the sum of all the horizontal forces exerted by the water is (by art. 58.) \( \frac{1}{2} sy^2 \), and the adhesion may be represented by some multiple m, of its weight, the constant the dyke quantity m being determined by experience. The adhesion will therefore be \( m \times \sigma \int x' y' \), and the equation of equilibrium will be

\[ \frac{1}{2} sy^2 = m \times \sigma \int x' y', \] the fluxion of which is \[ sy \dot{y} = m \times \sigma \times x' \dot{y}. \] Dividing by \( \dot{y} \) we have \[ sy = m \times \sigma \times x, \] and therefore \[ x : y = s : m \times n. \]

Hence the base BC of the dyke is to its altitude BA as the specific gravity of water is to a multiple m of the specific gravity of the dyke, m being a constant quantity which experiments alone can determine.

In a work by the Abbé Boifuit and M. Viallet, entitled Recherches sur la Construction la plus avantageuse des Digue, the reader will find a general solution of the preceding problem, in which the vertical efforts of the water and the adhesion of the surfaces are considered. This able work, which we have followed in the preceding investigation, contains much practical information on the construction of dykes of every kind; and may be considered as a continuation of the second part of Belidor's Architecture Hydraulique.

CHAP. II. Of Specific Gravities.

DEFINITION.

70. The absolute weights of different bodies of the same bulk are called their specific gravities or densities; and one body is said to be specifically heavier, or specifically lighter than another, when under the same bulk it contains a greater or less quantity of matter. Brats, for example, is said to have eight times the specific gravity of water, because one cubic inch of brats contains eight times the quantity of matter, or is eight times heavier than a cubic inch of water.

PROP. I.

71. Fluids pressing against each other in two or more communicating vessels, will be in equilibrium when the perpendicular altitudes above the level of their junction are in the inverse ratio of their specific gravities.

If a quantity of mercury be poured into the vessel FMN, it will be in equilibrium when it rises to the same level AHIB in both tubes. Take away an inch of mercury ACDH, and substitute in its room 13 1/2 inches of water FCDG. Then since mercury is 13 1/2 times heavier than water, 13 1/2 inches of water will have the same absolute weight as one inch of mercury, and the equilibrium will not be disturbed; for the column of water FD will exert the same pressure upon the surface CD of the mercury, as the smaller column of mercury did formerly. The surface of the mercury, therefore, will remain at IB: now, since AB, CE, are horizontal lines, AC will be equal to LK; but FC was made 13 1/2 times AC, therefore FC = 13 1/2 times IK, that is FC : IK = 13 1/2 : 1, the ratio between the specific gravities of mercury and water.

72. On this proposition depends the theory of the barometer. Let a quantity of mercury be introduced into the tube FMN, and let the pressure of the atmosphere be removed from the surface IB; the pressure of the air upon the other surface CD will be the same as if the tube FD were continued to the top of the atmosphere, and therefore, instead of the column of water FD we have a column of air equal to the height of the atmosphere acting against the mercury CDMIB; the mercury consequently will rise towards N, so that its height will be to the height of the atmosphere as the specific gravity of air is to the specific gravity of mercury; but as the density of the air diminishes as it recedes from the earth, we must take the specific gravity of the air at a mean height in the atmosphere. It is obvious from the proposition, that the altitude of the column of mercury which balances the column of air must be reckoned from CD the level of their junction; and that, when the specific gravity of the air is diminished, the mercury will fall, and will again rise when it regains its former density.

PROP. II.

73. If any body is immersed in a fluid, or floats on its surface, it is pressed upwards with a force equal to the weight of the quantity of fluid displaced.

Let mH be the section of a body immersed in the vessel AB filled with a fluid. Any portion mn of its upper surface is pressed downwards by the column of fluid Cm nD (43.); but the similar portion EF of its lower surface is pressed upwards with a column of fluid equal to CEFD, therefore the part EF is pressed up-wards with the difference of these forces, that is, with a force equivalent to the column of fluid mEFn, for Fig. 2. CEFD—Cm nD = mEFn. In the same way it may be shewn, that the remaining part FH is pressed upwards with a force equal to the weight of a column nFH o; and therefore it follows, that the rectangle mEH o is pressed upwards with a force equivalent to a column mEH o, that is, to the quantity of fluid displaced.

74. If the body floats in the fluid like CH in the When the vessel AB (fig. 3.) the same consequence will follow; parallelopiped floats for the body CH is evidently pressed upwards with a force equivalent to the column mEH o, that is, to the part immersed or the quantity of fluid displaced. Now as the fame may be demonstrated of every other section of a solid parallelopiped, we may conclude, that the proportion is true with respect to every solid whose section is rectangular.

75. When the solid has any other form as CD, however irregular, we may conceive its section to be divided into a number of very small rectangles no: then the small portion of the solid at n is pressed downwards by a column of particles mn, and the small portion at o is pressed upwards by a column of particles equal to no; therefore the difference of these forces, viz. the column no, is the force with which the portion o is pressed upwards. In the same manner it can be shewn, that every other similar portion of the lower surface of the solid CD is pressed upwards with a force equal to a column of particles whose height is equal to the vertical breadth of the solid; but all these columns of particles must occupy the same space as the solid itself, therefore any solid body immersed in a fluid, or floating on its surface, is pressed upwards with a force equal to the weight of the quantity of fluid displaced.

76. Cor. 1. When a body floats in a fluid, the weight of the quantity of fluid displaced is equal to the weight of the floating solid. For since the solid is in equilibrium with the fluid, the force which causes it equal to the weight of the quantity of fluid displaced; but the force which keeps a part of the solid immersed in the fluid is the weight of the solid, and the force which presses the solid upwards, and prevents it from sinking, is equivalent to the weight of the quantity of fluid displaced (73.); therefore these forces and the weights to which they are equivalent must be equal.

77. Cor. 2. A solid weighed in a fluid loses as much of its weight as is equal to the weight of the quantity of fluid displaced; for since the body is pressed upwards with a force equal to the weight of the fluid displaced (3.), this pressure acts in direct opposition to the natural gravity or absolute weight of the solid, and therefore diminishes its absolute weight by a quantity equal to the weight of the fluid displaced. The part of the weight thus lost is not destroyed: Is is only sustained by a force acting in a contrary direction.

78. Cor. 3. A solid immersed in a fluid will sink, if its specific gravity exceed that of the fluid: It will float Of Specific float on the surface, partly immersed, if its specific gravity be less than that of the fluid; and it will remain wholly immersed wherever it is placed, if the specific gravities of the solid and fluid are equal. In the first case, the force with which the solid is pressed downwardly exceeds the upward pressure, and therefore it must sink. In the second case, the upward pressure exceeds the pressure downwards, and therefore the body must float; and, in the third case, the upward and downward pressures being equal, the solid will remain wherever it is placed.

79. Cor. 4. The specific gravities of two or more fluids are to one another as the losses of weight sustained by the same solid body, and specifically heavier than the fluids, when weighed in each fluid respectively. The solid in this case displaces equal quantities of each fluid; but the losses of weight are respectively as the absolute weights of the quantities displaced (Cor. 2.), therefore the specific gravities, which are as the absolute weights of equal quantities of any body (70.), must be as the losses of weight sustained by the immersed solid.

80. Cor. 5. The specific gravity of a solid is to that of a fluid as the absolute weight of the solid is to the loss of weight which it sustains when weighed in the fluid. For since the loss of weight sustained by the solid is equal to the absolute weight of the quantity of fluid displaced, or of a quantity of fluid of the same bulk as the solid, the specific gravities, which (70.) are in the ratio of the absolute weights of equal volumes, must be as the absolute weight of the solid to the loss weight which it sustains.

81. Cor. 6. The specific gravity of a solid floating in a fluid, is to the specific gravity of the fluid itself, as the bulk of the part immersed is to the total bulk of the solid.

82. Cor. 7. Bodies which sustain equal losses of weight are of the same bulk. For, since the losses of weight are as the weights of the quantities of fluid displaced, and as the quantities displaced are as the bulks of the solids which displace them, the bulks must be equal when the losses of weight are equal.

The preceding corollaries may be expressed algebraically, and may be deduced from a general equation in the following manner. Let B be the total bulk of a floating body, and C the part of it which is immersed; let S be the specific gravity of the solid, and s that of the fluid. Then it is obvious, that the absolute weight of the solid will be expressed by \( B \times S \), and the absolute weight of the fluid displaced by \( C \times s \); for the fluid displaced has the same bulk as the part of the solid which is immersed. In order that an equilibrium may obtain between the solid and fluid, we must have \( B \times S = C \times s \): Now, when \( s > S \), we have \( B > C \), so that the solid will float, which is the second case of Cor. 3.—When \( S = s \) we have \( B = C \), which is the third case of Cor. 3.—When \( S < s \) we have \( C > B \), that is, the body will sink below the surface; and it will descend to the bottom, for it cannot be suspended in the fluid without some power to support it; and if such a power were necessary, we should have \( B \times S > C \times s \), which is contrary to the equation of equilibrium.

84. From the equation \( B \times S = C \times s \) we have (Euclid VI. 16.) \( S : s = B : C \), which is Cor. 6.—When the body is completely immersed we have \( B = C \), in which case the equation becomes \( B \times S = B \times s \); and when the solid is specifically heavier than the fluid, it will require a counterweight to keep the solid suspended in the fluid. Let W be the counterweight necessary for keeping the solid suspended in the fluid, then in the case of an equilibrium the equation will be \( B \times s + W = B \times S \), or \( B \times S - W = B \times s \), or \( S \times B \times S - W = S \times B \times s \), whence (Euclid VI. 16.) \( S : s = B \times S : B \times S - W \), which is Cor. 5.

85. If the same solid body is plunged in a second fluid of a different specific gravity from the first, let \( \sigma \) be the specific gravity of the second fluid, and w the counterweight necessary to keep the solid suspended in it. The equation for the first fluid was \( B \times s + W = B \times S \) (84.), and the equation for the second fluid will be \( B \times s + w = B \times S \); therefore we shall have, by the first equation, \( S \times B - W = s \times B \), and by the second \( S \times B - w = \sigma \times B \), and consequently \( s \times B : \sigma \times B = S \times B - W : S \times B - w \), or (Euclid V. 16.) \( s : \sigma = S \times B - W : S \times B - w \), which is Cor. 4.; for the losses of weight in each fluid are evidently represented by \( S \times B - W \) and \( S \times B - w \).

86. If B and b express the bulks of two solids, S and s their specific gravities, \( \sigma \) the specific gravity of the fluid, and W, w the counterweights which keep them in equilibrium with the fluid. Then with the solid S the equation will be \( S \times B - W = s \times B \) (85.); and with the solid s the equation will be \( s \times b - w = \sigma \times b \). Wherefore, if the two solids sustain equal losses of weight, we shall have \( S \times B - W = s \times b - w \), since each side of the equation represents the loss of weight sustained by each solid respectively. Consequently, \( \sigma \times B = s \times b \), and dividing by \( \sigma \), we have \( B = b \), which is corollary 7.

87. From the preceding proposition and its corollaries, we may deduce a method of detecting adulteration in the precious metals, and of revolting the problem proposed to Archimedes, by Hiero king of Syracuse. Take a real guinea, and a counterfeit one made of copper and gold. If the latter be lighter than the former, when weighed in a pair of scales, the impostion is instantly detected: But should their weight be the same, let the two coins be weighed in water, and let the loss of weight sustained by each be carefully observed, it will then be found that the counterfeit will lose more of its weight than the unadulterated coin. For, since the specific gravity of copper exceeds that of gold, and since the absolute weights of the coins were equal, the counterfeit guinea must be greater in bulk than the real one, and will therefore displace a greater quantity of water, that is (77.), it will lose a greater part of its weight.

88. Hiero, king of Syracuse, having employed a goldsmith to make him a crown of gold, suspected that the metal had been adulterated, and inquired at Archimedes if his suspicions could be verified or disproved without injuring the crown. The particular method by which Archimedes detected the fraud of the goldsmith is not certainly known; but it is probable that he did it in the following manner. A quantity of gold, of the same absolute weight as the crown, would evidently have the same bulk also, if the crown were pure gold, and would have a greater bulk if the crown were made

Of Specific made of adulterated gold. By weighing, therefore, Gravities. the quantity of gold and the crown in water, and observing their respective losses of weight, Archimedes found that the crown lost more of its weight than the quantity of gold; and therefore concluded, that as the crown must have displaced a greater portion of water than the piece of gold, its bulk must likewise have been greater, and the metal adulterated of which it was composed.

PROF. III.

39. If two immiscible fluids, of different specific gravities, and a solid of an intermediate specific gravity, be put into a vessel, the part of the solid in the lighter fluid will be to the whole solid, as the difference between the specific gravities of the solid and the heavier fluid, is to the difference between the specific gravities of the two fluids.

Let AB (fig. 5.) be the vessel which contains the two fluids, suppose mercury and water, and the solid CD. The mercury being heavier than water will sink to the bottom and have m n for its surface, and the water will occupy the space AB m n. The solid having a greater specific gravity than water, will sink in the water (78.); but having a less specific gravity than mercury, it will float in the mercury. It will, therefore, be suspended in the fluids, having one portion C in the water, and the other portion D in the mercury. Now let S be the specific gravity of the mercury, s the specific gravity of the water, σ that of the solid, C the part of the solid in the water, and D the part in the mercury. Then the bulk of the solid is C + D, and its weight \( \sigma \times C + D \): The quantity of water displaced by the part C, or the loss of weight sustained by the part C, will be \( C \times s \); and the quantity of mercury displaced, or the loss of weight sustained by part D, will be \( D \times S \). But as the solid is suspended in the fluids, and therefore in equilibrium with them, the whole of its weight is lost. Consequently, the part of its weight which is lost in the water, added to the part lost in the mercury, must be equal to its whole weight, that is, \( C \times s + D \times S = \sigma \times C + D \), or \( sC + SD = \sigma C + \sigma D \). Transposing \( \sigma C \) and \( \sigma D \), we have \( sC - \sigma C = SD - \sigma D \), or \( C \times s - \sigma = D \times S - \sigma \), and (Euclid VI. 16.) \( C : D = \sigma : S - \sigma \). Then, by inversion and composition (Euclid V. Propositions B and 18.) \( C : C + D = S - \sigma : S - \sigma \). Q. E. D.

90. Cor. 1. From the analogy \( C : D = \sigma : S - \sigma \), we learn that the part of the solid in the heavier fluid, is to the part in the lighter fluid, as the difference between the specific gravities of the solid and the lighter fluid, is to the difference between the specific gravities of the solid and the heavier fluid.

91. Cor. 2. When \( s \) is very small compared with \( S \), we may use the analogy \( C : C + D = \sigma : s \), though in cases where great accuracy is necessary this ought not to be done. When the specific gravity of a body, lighter than water, is determined by comparing the part immersed with the whole body, there is evidently a small error in the result; for the body is suspended partly in water and partly in air. It is in fact a solid of an intermediate specific gravity floating in two immiscible fluids, and therefore its specific gravity should be ascertained by the present proposition.

PROP. IV.

92. If two bodies, whether solid or fluid, be mixed together so as to form a compound substance, the bulk of the heavier is to the bulk of the lighter ingredient, as the difference between the specific gravities of the compound, and the lighter ingredient, is to the difference between the specific gravities of the compound and the heavier ingredient.

Let S and s be the specific gravities of the two ingredients, the specific gravity of the compound, and B, b the bulks of the ingredients; then the bulk of the compound will be \( B + b \), and its weight \( \sigma \times B + b \). The weight of the ingredient B will be \( B \times S \), and that of the other ingredient \( b \times s \); and as the weight of the compound must be equal to the weight of its ingredients, we have the following equation. \( \sigma \times B + b \times s = B \times S + b \times s \), and by transposing \( \sigma \times B \) and BS, we shall have \( B \times S - BS = b \times s - b \times s \), or \( B \times S - S = b \times s - s \); therefore (Euclid VI. 16.) \( B : b = S : s \). Q. E. D.

93. In the preceding proposition, it has been taken for granted that the magnitude of the compound is exactly equal to the sum of the magnitudes of the two ingredients. This, however, does not obtain universally either in fluids or solids; for an increase or diminution of bulk often attends the combination of two different ingredients. A cubical inch of alcohol, for example, combined with a cubical inch of water, will form a compound which will measure less than two cubic inches; and a cubical inch of tin, when incorporated in a fluid state with a cubical inch of lead, will form a compound, whose bulk will exceed two cubical inches. The preceding proposition, however, is, even in these cases, of great use in ascertaining the increase or decrease of bulk sustained by the compound, by comparing the computed with the observed bulk. See SPECIFIC Gravity.

PROP. V. PROBLEM.

94. How to determine the specific gravities of bodies whether solid or fluid.

The simplest and most natural way of finding the specific gravities of bodies would be to take the absolute weights of a cubic inch, or any other determinate quantity, of each substance; and the number thus found would be their specific gravities. But as it is difficult to form two bodies of the very same size, and often impossible, as in the case of precious stones, to give a determinate form to the substance under examination, we are obliged to weigh them in a fluid, and deduce their specific gravities from the losses of weight which they generally sustain. Water is a fluid which is always employed for this purpose, not only because it can be had without difficulty, but because it can be procured of the same temperature, and of the same density, in every part of the world. The specific gravity of water is always called 1.000, and with this, as a standard, the specific gravity of every other substance is compared. Thus, if

Of Specific Gravities. Of a certain quantity of water weighed four pounds, and a similar quantity of mercury 56 pounds, the specific gravity of the mercury would be called 14, because as 4 : 56 = 1 : 14. In order, therefore, to determine the densities of bodies, we have occasion for no other instrument than a common balance with a hook fixed beneath one of its scales. When fitted up in this way, it has been called the hydrostatic balance, which has already been described under the article BALANCE, Hydrostatical.

To find the specific gravity of a solid heavier than water.—Suspend the solid by means of a fine silver wire to the hook beneath the scale, and find its weight in air. Fill a jar with pure distilled water, of the temperature of 62° of Fahrenheit's thermometer, and find the weight of the solid when immersed in this fluid. The difference of these weights is the losf of weight sustained by the solid. Then, (82.) as the losf of weight is to the weight of the solid in air, so is 1.000 the specific gravity of water to a fourth proportional, which will be the specific gravity of the solid. But as the third term of the preceding analogy is always 1.000, the fourth proportional, or density of the solid, will always be had by dividing the weight of the solid in air by its losf of weight in water. If the solid substance consists of grains of platina or metallic filings, place it in a small glass bucket. Find the weight of the bucket in air, when empty, and also its weight when it contains the substance. The difference of these weights will be the weight of the substance in air. Do the very same in water, and its weight in water will be had. Its specific gravity will then be found as formerly.—If the body is soluble in water, or so porous as to absorb it, it should be covered with varnish or some unctuous substance. When it is weighed in water, it should never touch the sides of the glass jar, and it must be carefully freed from any bubbles of air that happen to adhere to it.

To find the specific gravity of a solid lighter than water.—Fasten to it another solid heavier than water, so that they may sink together. Find the weight of the denser body, and also of the compound body, both in air and in water; and by subtracting their weight in water from their weight in air, find how much weight they have severally lost. Then say as the difference between their losfs of weight is to the weight of the light body in air, so is 1.000 to the specific gravity of the body.

To find the specific gravity of powders. 97. When the substance is a powder which absorbs water, or is soluble in it.—Place a glass phial in one scale, and counterpoise it by weights in the other. Fill this phial with the powder to be examined; and having rammed it as close as possible to the very top, find the weight of the powder. Remove the powder from the phial, and fill it with distilled water, and find its weight. The weight of the powder, divided by the weight of the water, will be the specific gravity of the former.

To find the specific gravity of fluids. 98. When the substance is a fluid, its specific gravity may be determined very accurately by the method in the preceding article, or by the following method deduced from article 79.—Take any solid specifically heavier than water, and the given fluid. Find the losf of weight which it sustains in water, and also in the given fluid. Then, since the specific gravities are as the losfs of weight sustained by the same solid, the specific gravity of the fluid required will be found by dividing the losf of weight sustained by the solid in the given fluid, by the losf of weight which it sustains in water.

SECT. II. On the Hydrometer.

99. In order to determine, with expedition, the strength of spirituous liquors, which are inversely proportional to their specific gravities, an instrument more simple and invented by Hypathia, 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.

100. The hydrometer of Fahrenheit, which is one of the simplest that has been constructed, is represented in heit's fig. 6, and may be formed either of glass or metal. AB is a cylindrical stem, and C, D two hollow balls Fig. 6. appended to it. Into the lower ball D is introduced a quantity of mercury, sufficient to make the ball C sink to F, a little below the surface of distilled water. If this apparatus be plunged into a fluid lighter than water, the ball C will sink farther below the surface; and if it be immersed in a heavier fluid, it will rise nearer the surface. In this way we can tell whether one fluid is more or less dense than another. But in order to determine the real specific gravities of the fluids, the hydrometer must either be loaded with different weights, or have a scale AB engraven on its stem. The former of these methods was employed by Fahrenheit. Having placed some small weights on the top A, he marked any point E, to which the instrument sunk in distilled water. By weighing the instrument thus loaded, he found the weight of a quantity of water equal to the part immersed (76.). When the hydrometer was placed in a fluid denser than water, he loaded it with additional weights till it sunk to the same point E. The Hydrome-weight of the hydrometer being again found, gave him ter with the weight of a quantity of the denser fluid equal to the weights part immersed; but as the part immersed was the same in both cases, the weights of the hydrometer were equal to the absolute weights of equal quantities of the two fluids; and consequently the specific gravities of the water and the other fluid were in the ratio of these weights. When the fluid, whose density is required, has less specific gravity than water, some of the weights are to be removed from the top A till the instrument sinks to E; and the density of the fluid to be determined as before.—Instead of making the weight of Hydrome-the hydrometer variable, it is more simple, though less accurate, to have a scale of equal parts upon the engraved stem AB. In order to graduate this scale, immerse the hydrometer in distilled water, at the temperature of 60° Fahrenheit, so that it may sink to B near the bottom of the stem, which may be easily effected, by diminishing or increasing the quantity of mercury in the ball D. At B place the number 1.000, which shews that every fluid, in which the hydrometer sinks to B, has its specific gravity 1.000, or that of distilled water. The hydrometer is then to be plunged in another fluid less dense than water, suppose oil, whose specific gravity Chap. II.

Of Specific Gravities, it sinks. Every fluid, therefore, in which the hydrometer sinks to A, has its specific gravity .900; and if the scale AB be divided into equal parts, every intermediate degree of specific gravity between .900 and 1.000 will be marked. If the scale AB be divided into four parts in the points E, F, G, the fluid in which the hydrometer sinks to G will have .975 for its specific gravity; the specific gravity of that in which it sinks to F will be .950, and so on with the other points of division. If it is required to extend the range of the instrument, and to make it indicate the densities of fluids specifically lighter than water, we have only to load it in such a manner as to make it sink to the middle of the scale F in distilled water; and by taking two fluids, between whose densities the specific gravity of every other fluid is contained, excepting mercury and metals in a fluid state, to determine, as before, the extremities of the scale.

101. When the weight of the hydrometer is variable, let E be the point to which it sinks in two different fluids; and let W be the absolute weight necessary to make it sink to E in the denser fluid, and \( \frac{W}{s} \) the weight necessary to make it sink to the same point in the lighter fluid. Let S, s be the specific gravities of the two fluids, and V the volume of the part of the hydrometer that is constantly immered. Then (83.) \( W = S \times V, \quad \frac{W}{s} = s \times V \). From the first equation we have, \( V = \frac{W}{S} \), and from the second equation \( V = \frac{W}{s} \), consequently \( \frac{W}{s} = \frac{W}{S} \), and by reduction \( s = \frac{s \times W}{W} \). Thus, by knowing W and the weight p, and also S the specific gravity of one of the fluids, which will be 1.000 if that fluid be water, we can find s the specific gravity of the other fluid.

102. When the weight of the hydrometer is constant, and the density of the fluid indicated by the depth to which it descends, let F, E be the points to which it sinks in two different fluids, whose specific gravities are S, s, W the absolute weight of the hydrometer, V the volume of the part immered when the hydrometer has sunk to E, and v its volume when sunk to F. Then (83.), we have \( W = S \times V \), and \( W = s \times v \), consequently \( s \times v = S \times V \), and \( s = \frac{S \times V}{v} \). If the absolute weight W, therefore, of the hydrometer be known, and also the volumes V, v, and the specific gravity S of one of the fluids, which may be water, the specific gravity of the other fluid may be determined by the preceding formula. When the figure of the hydrometer is regular, the volumes V, v, may be determined geometrically; but as the instrument is generally of an irregular form, the following method should be employed.

103. The hydrometers of Clarke and Desaguliers differ so little from those which have now been described, that they are not entitled to a more particular description. The hydrometer invented by Mr William Jones of Holborn, is a simple and accurate instrument, and requires only three weights to discover the strengths of spirituous liquors from alcohol to water. Like other instruments of the same kind, it is adjusted to the temperature of 60° of Fahrenheit; but as every change of temperature produces a change in the specific gravity of the spirits, Mr Jones found it necessary to attach a thermometer to the instrument, and thus make a proper allowance for every variation of temperature. Almost all bodies expand with heat and contract with cold; and as their volume becomes different at different temperatures, their specific gravities must also (70.) be variable, and will diminish with an increase of temperature. M. Homberg, and M. Eiffenclhmed found that the absolute weight of a cubic inch of brandy was four drams 42 grains in winter, and only four drams 32 grains in summer, and that the difference in spirits of nitre was still greater. It has been found, indeed, upon an average, that 32 gallons of spirits in winter will expand to 33 gallons in summer. As the strength of spirituous liquors is inversely as their specific gravities, they will appear much stronger in summer than in winter. This change in their strength had been formerly estimated in a rough way; but by the application of the thermometer, and by adjusting its divisions experimentally, Mr Jones has reduced it to pretty accurate computation. It has already been stated (93.) that where two substances are combined, the magnitude of the compound body is sometimes greater and sometimes less than the sum of the magnitudes of the two ingredients, and that this mutual penetration particularly happened in the mixture of alcohol and water. In strong spirits, this concentration is sometimes so great, as to produce a diminution of four gallons in the 100; for if to 100 gallons of spirit of wine found by the hydrometer to be 66 gallons in the 100 over proof, you add 66 gallons of water to reduce it to proof, the mixture will consist only of 162 gallons instead of 166 of proof spirits. This mutual penetration of the particles of alcohol and water has also been considered in Mr Jones's hydrometer, which we shall now describe with greater minuteness.

104. In fig. 7. 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 brafs, and nearly oval, having its conjugate diameter about 1½ inches. The item AD is a parallelopiped, on the four sides of which the different strengths of spirits are engraved: the three sides which do not appear in fig. 7. are represented in fig. 8. 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 shewn on the side AD marked o at the top, and any degree of strength from 74 gallons in the 100 to 47 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 immered. If the weight No 1 is necessary, the side marked 1 will show the strength of the 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 overproof 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, 3½ at 61, 2½ at 48 (fig. 7.) indicate the diminution

Of Specific of bulk which takes place when water is mixed with Gravities. 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 37 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 o, 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 o, and is equivalent to a temperature of 60 degrees 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 180 must the spirit be reckoned stronger.

Dicas's hydrometer with a sliding rule.

105. The patent hydrometer invented by Mr Dicas of Liverpool, possesses all the advantages of that which has now been described, but is superior to it in regard to the accuracy with which it estimates the aberration arising from a change of temperature. It is constructed in the common form, with 36 different weights, which are valued from o to 370, including the divisions on the stem; but the chief improvement consists in an ivory sliding rule which accompanies the instrument. In order to understand the construction of this sliding rule the reader must have recourse to the instrument itself.

Quin's universal hydrometer.

106. Quin's universal hydrometer is constructed in such a manner, as to ascertain, with the greatest expedition, the strength of any spirit from alcohol to water, and also the concentration and specific gravity of each different strength. With the assistance of four weights, it discovers likewise the gravity of worts, and is therefore of more universal use than any other hydrometer. The instrument is represented in fig. 9, with the four sides of its stem graduated and marked at the top so as to correspond with the weights below. The side of the stem marked A, B, C, D, &c. to Z, shows the strength of any spirit from alcohol to water, and the three other sides numbered 1, 2, 3, are adapted for worts. The variation of density arising from the contraction and dilatation of the fluid is determined by means of a sliding rule, differing very little from that of Mr Dicas. In order to use this instrument, place any of the weights, if necessary, on the stem at C; find the temperature of the spirit by a thermometer, and bring the star on the sliding rule to the degree of heat on the thermometer's scale: then opposite to the number of the weight and the letter on the stem, you have the strength of the spirit pointed out on the sliding rule, which is lettered and numbered in the same way as the instrument and weights. In ascertaining the strength of worts, the weight No 4 is always to continue on the hydrometer, and the weights, No 1, 2, 3, are adapted to the sides No 1, 2, 3, of the square stem, which point out the exact gravity of the worts.

Nicholson's hydrometer.

107. A considerable improvement on the hydrometer has lately been made by Mr Nicholson, who has rendered it capable of ascertaining the specific gravities both of solids and fluids. In fig. 10. F is a hollow ball of copper attached to the dish AA by a stem B, made of hardened steel. To the lower extremity of the ball is affixed a kind of iron stirrup FF, carrying another dish G of such a weight as to keep the stem vertical when the instrument is afloat. The parts of the hydrometer are so adjusted, that when the lower dish G is empty, and the upper dish AA contains 1000 grains, it will sink in distilled water at the temperature of 60° of Fahrenheit, so that the surface of the fluid may cut the stem DB at the point D. In order to measure the specific gravities of fluids, let the weight of the instrument, when loaded, be accurately ascertained. Then, this weight is equal to that of a quantity of distilled water at the temperature of 60°, having the same volume as that part of the instrument which is below the point D of the stem. If the hydrometer, therefore, is immersed to the point D in any other fluid of the same temperature, which may be done by increasing or diminishing the weights in the dish AA, the difference between this last weight and 1000 grains will express the difference between equal bulks of water and the other fluid. Now as the weight of the mass of water is equal to the weight of the instrument, which may be called W, the above-mentioned difference or D must be either added to or subtracted from W, (according as the weight in the dish AA was increased or diminished) in order to have the weight of an equal bulk of the fluid; then \( \frac{W \pm D}{W} \) will be to W as the specific gravity of the given fluid is to that of water. This ratio will be expressed with considerable accuracy, as the cylindrical stem of the instrument being no more than \( \frac{1}{25} \)th of an inch in diameter, will be elevated or depressed nearly an inch by the subtraction or addition of \( \frac{1}{10} \) of a grain, and will, therefore, easily point out any changes of weight, not less than \( \frac{1}{10} \) of a grain, or \( \frac{1}{25000} \) of the whole, which will give the specific gravities to five places of figures. The solid bodies whose specific gravities are to be determined by this hydrometer, must not exceed 1000 grains in weight. For this purpose, immerse the instrument in distilled water, and load the upper dish till the surface of the water is on a level with the point D of the stem. Then, if the weight required to produce this equilibrium be exactly 1000 grains, the temperature of the water will be 60° of Fahrenheit; but if they be greater or less than 1000 grains, the water will be colder or warmer. After noting down the weight necessary for producing an equilibrium, unload the upper dish, and place on it the body whose specific gravity is required. Increase the weight in the upper dish, till the instrument sinks to the point D, and the difference between this new weight and the weight formerly noted down will be the weight of the body in air. Place the body in the lower dish G, and add weights in the upper dish till the hydrometer again sinks to D. This weight will be the difference between 1000 grains and the weight of the body in water; and since the weight of the body in air, and its weight in water, are ascertained, its loss of weight will be known, and consequently its specific gravity (8o.).

De Parcieux's areometer.

108. The areometer or hydrometer of M. De Parcieux consists of a small glass phial EG, about two inches in diameter

Of Specific diameter and seven inches long, having its bottom as flat as possible. The mouth is closed with a cork stopper, into which is inserted a straight iron or bra's wire EF, about a line in diameter, and 30 inches long.

When two fluids are to be compared, the bottle is loaded in such a manner by the introduction of small shot, that the instrument, when plunged in the lightest of the fluids, sinks so deep as to leave only the extremity of the wire above its surface, while in the heaviest fluid, the wire is some inches below the surface. The same effect may be produced by fixing a little dish F to the top of the wire, and varying the weights, or by altering the thickness of the wire. The areometer thus constructed, will indicate the smallest differences of specific gravity, and such minute variations of density, arising from a change of temperature, which would be imperceptible by any other hydrometer. The motion of an instrument of this kind, says Montucla, was so sensible, that when immersed in water of the usual temperature, it sunk several inches while the rays of the sun fell upon the water, and instantly rose when his rays were intercepted. In one of the areometers used by Deparcieux, an interval of fix lines in the stem corresponded to a change of density about \( \frac{1}{75} \) of the whole. (Mem. de l'Acad. Paris 1766, p. 138.).

In order to determine the strength of spirits with the greatest expedition, Professor Wilfon of Glasgow employed a very simple method. His hydrometer consists of a number of glass beads, the specific gravities of each of which vary in a known ratio. When the strength of any spirit is to be tried, the glass beads, which are all numbered, are to be thrown into it. Some of those whose specific gravity exceeds that of the spirit, will sink to the bottom, while others will swim on the top, or remain suspended in the fluid. That which neither sinks to the bottom nor swims on the surface, will indicate by its number the specific gravity of the spirits (78.). For an account of the Patent Beads constructed by Mrs Lovi of Edinburgh, in all respects the most accurate apparatus for this purpose, see note under SPIRITUOUS LIQUORS.

SECT. III. On Tables of Specific Gravities.

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, published in the enlarged edition of Ferguson's Lectures, 2d edition. It comprehends the greater part of Briffon's tables, and is one of the most extensive that has yet been published. The names of the minerals, as given in Kirwan's Mineralogy, have in general been adopted; and such as have been discovered since the publication of that work will be found in later works. When the specific gravities of any substance, as determined by different authors, seem to be at variance, the different results are frequently given, and the names of the chemists prefixed by whom these results were obtained. The substances in the table have, contrary to the usual practice, been disposed in an alphabetical order. This was deemed more convenient for the purposes of reference, than if they had been divided into classes, or arranged according to the order of their densities.

<table> <tr> <th>A</th> <th></th> <th></th> </tr> <tr> <td>ACACIA, inspissated juice of,</td> <td>1.5133</td> <td></td> </tr> <tr> <td>Acid, nitric,</td> <td>1.2715</td> <td></td> </tr> <tr> <td>muriatic,</td> <td>1.1940</td> <td></td> </tr> <tr> <td>red acetous,</td> <td>1.0251</td> <td></td> </tr> <tr> <td>white acetous,</td> <td>1.0135</td> <td></td> </tr> <tr> <td>distilled acetous,</td> <td>1.0095</td> <td></td> </tr> <tr> <td>acetic,</td> <td>1.0626</td> <td></td> </tr> <tr> <td>sulphuric,</td> <td>1.8409</td> <td></td> </tr> <tr> <td>highly concentrated,</td> <td>2.125</td> <td></td> </tr> <tr> <td>nitric, highly concentrated,</td> <td>1.580</td> <td></td> </tr> <tr> <td>fluoric,</td> <td>1.500</td> <td></td> </tr> <tr> <td>formic,</td> <td>0.9942</td> <td></td> </tr> <tr> <td>phosphoric,</td> <td>1.5575</td> <td></td> </tr> <tr> <td>citric,</td> <td>1.0345</td> <td></td> </tr> <tr> <td>arsenic,</td> <td>1.8731</td> <td></td> </tr> <tr> <td>of oranges,</td> <td>1.0176</td> <td></td> </tr> <tr> <td>of gooseberries,</td> <td>1.0581</td> <td></td> </tr> <tr> <td>of grapes,</td> <td>1.0241</td> <td></td> </tr> <tr> <td>Agynolite, glaify,</td> <td>Kirwan.</td> <td>2.950<br>3.903</td> <td></td> </tr> <tr> <td>Æther, sulphuric,</td> <td></td> <td>0.7396</td> <td></td> </tr> <tr> <td>nitric,</td> <td></td> <td>0.9088</td> <td></td> </tr> <tr> <td>muriatic,</td> <td></td> <td>0.7206</td> <td></td> </tr> <tr> <td>acetic,</td> <td></td> <td>0.8664</td> <td></td> </tr> <tr> <td>Agate, oriental,</td> <td></td> <td>0.5901</td> <td></td> </tr> <tr> <td>onyx,</td> <td></td> <td>2.6375</td> <td></td> </tr> <tr> <td>speckled,</td> <td></td> <td>2.607</td> <td></td> </tr> <tr> <td>cloudy,</td> <td></td> <td>2.6253</td> <td></td> </tr> <tr> <td>Agate, stained;</td> <td></td> <td>2.6324</td> <td></td> </tr> <tr> <td>veined,</td> <td></td> <td>2.6667</td> <td></td> </tr> <tr> <td>Icelandic,</td> <td></td> <td>2.348</td> <td></td> </tr> <tr> <td>of Havre,</td> <td></td> <td>2.5881</td> <td></td> </tr> <tr> <td>Jafpee,</td> <td></td> <td>2.6356</td> <td></td> </tr> <tr> <td>Herborisée,</td> <td></td> <td>2.5891</td> <td></td> </tr> <tr> <td>Irisée,</td> <td></td> <td>2.5535</td> <td></td> </tr> <tr> <td>Air, atmospheric,</td> <td></td> <td>Barom. 29.75<br>Thermom. 32.<br>Barom. 29.85<br>Thermom. 54°.5</td> <td>Lavoisier. 0.00122<br>0.0012308</td> </tr> <tr> <td>Alabaster of Valencia,</td> <td></td> <td>veined,<br>of Piedmont,<br>of Malta,<br>yellow,<br>Spanish saline,<br>oriental white,<br>ditto, femitransparent,<br>stained brown,<br>of Malaga pink,<br>of Dalias,</td> <td>2.638<br>2.691<br>2.693<br>2.699<br>2.699<br>2.713<br>2.730<br>2.762<br>2.744<br>2.8761<br>2.6110</td> </tr> <tr> <td>Alcohol, highly rectified,</td> <td></td> <td>commercial,<br>15 parts water 1 part</td> <td>0.8293<br>0.7371<br>0.8527</td> </tr> <tr> <td></td> <td></td> <td>14<br>13<br>12<br>11</td> <td>0.8674<br>0.8815<br>0.8947<br>0.9075</td> </tr> <tr> <td></td> <td></td> <td></td> <td>Alcohol,</td> </tr> </table>

Of Specific Alcohol, 10 parts water 6 Gravities. 9 7 8 8 7 9 6 10 5 11 4 12 3 13 2 14 1 15

Alder wood, Aloes, hepatic, focotrine, Alouchi, odoriferous gum, Alumine, sulphate of, saturated solution of, temp. 42°, Watfon. Amber, yellow transparent, red, green, Ambergris, Amethyst, common. See Rock crystal. Amianthus, long, penetrated with water, short, penetrated with water, Amianthinite from Raichau, Bayreuth, Ammoniac, liquid, muriate of, saturated solution of, temp. 42°, Watfon. Andalusite, or hardspar, Anime, oriental, occidental, Antimony, glass of, in a metallic state, fused, native, sulphur of, Antimonial ore, gray and foliated, radiated, red, Apple tree, Aquamarine. See Beryl. Arcanfon, Areca, inspissated juice of, Arctizite, or wernerite, Argillite, or flate clay, Arnotto, Arragon spar, Arsenic bloom, Pharmacolite, fused, native, pyrites, common, native, orpiment, glass of, (arsenic of the shops), Asbestinite,

0.9199 0.9317 0.9427 0.9519 0.9594 0.9674 0.9733 0.9791 0.9852 0.9919 0.8000 1.3868 1.3795 1.0604 1.7140 1.033 1.0780 1.0855 1.0834 1.0829 0.7800 0.9263 2.750 0.9088 1.5662 2.3134 3.3803 2.584 2.916 0.8970 1.4530 1.072 3.165 1.0284 1.0426 4.9464 6.624 6.860 6.720 4.0643 4.368 4.440 3.750 4.090 0.7930 1.0857 1.4573 3.606 2.600 2.680 0.5956 2.946 2.640 8.310 5.670 4.791 5.600 6.522 5.452 3.5942 3.000 3.310 0.9317 0.9427 0.9519 0.9594 0.9674 0.9733 0.9791 0.9852 0.9919 0.8000 1.3868 1.3795 1.0604 1.7140 1.033 1.0780 1.0855 1.0834 1.0829 0.7800 0.9263 2.750 0.9088 1.5662 2.3134 3.3803 2.584 2.916 0.8970 1.4530 1.072 3.165 1.0284 1.0426 4.9464 6.624 6.860 6.720 4.0643 4.368 4.440 3.750 4.090 0.7930 1.0857 1.4573 3.606 2.600 2.680 0.5956 2.946 2.640 8.310 5.670 4.791 5.600 6.522 5.452 3.5942 3.000 3.310 Asbestos, mountain cork, penetrated with water, ripe, penetrated with water, flarry, penetrated with water, unripe, penetrated with water, Ash trunk, dry, Asphaltum, cohesive, compact, Asafetida, Aventurine, semitransparent, opaque, Augite, octahedral basaltes, oriental, of Siberia, Azure stone, or lapis lazuli, oriental, Barolite, or witherite, Baroselenite, or barytes, white, grey, rhomboidal, octahedral, in stalactites, sulphate of, native, carbonate of, native, Bafaltes, from the Giant's causeway, prismatic from Auvergne, of St Tubery, Barsas, a juice of the pine, Bay tree, Spanish, Bdellium, Beech-wood, Beer, red, white, Benzoin, Beryl, oriental aquamarine, occidental, or aquamarine, schorlous, or shorlite, Bezoar, oriental, occidental, Bismuth, native, sulphurated, ochre, in a metallic state, fused, Bergman. { 0.6806 Of Specific Gravities. { 0.9933 { 1.2492 { 1.3492 Briffon. 2.5779 2.6994 3.0733 3.0808 2.9958 3.0343 Muschchenbroek. 0.8450 Turin. 0.800 { 1.450 { 2.060 { 1.070 { 1.165 Häuy. 3.226 Werner. 3.471 Reyss. 3.777 Briffon. 2.7675 Kirwan. 2.896 2.7714 2.9454 { 4.300 { 4.338 { 4.400 { 4.865 { 4.4300 { 4.4999 { 4.4434 { 4.4712 { 4.2984 { 4.000 { 4.460 { 4.300 { 4.338 { 2.979 { 3.000 { 2.864 { 2.4215 { 2.7948 { 1.0441 { 0.8220 { 1.1377 { 0.8520 { 1.0338 { 1.0231 { 1.0924 { 3.5491 { 2.723 { 2.650 { 2.759 { 3.514 { 1.666 { 2.233 { 9.570 { 6.131 { 4.371 { 9.756 { 9.822 Bitumen,

Bitumen, of Judea, 1.104 Black-coal, pitch coal, 1.308 slate coal, English, Kirwan. 1.250 Bielschowitz, Richter. 1.370 cannel coal, La Metherie. 1.321 Blende, yellow, Gellert. 1.382 brown, foliated, Gellert. 4.044 black, Gellert. 4.048 auriferous from Nag. Gellert. 3.770 yag, Von Muller. 4.048 Blood, human, Jurin. 3.930 craftimentum of, Jurin. 2.664 ferum of, Jurin. 2.664 Boles, Bone of an ox, Kirwan. 2.664 Boracite, Weßtrumb. 2.664 Borax, saturated solution of, temp. 42°, Watson. 2.664 Bournonite, 5.576 Boxwood, French, Muschenbroek. 0.9120 Dutch, Muschenbroek. 1.3280 dry, Jurin. 1.030 Bras, common, cast, 7.824 wiredrawn, 8.544 cast, not hammered, 8.396 Brazil wood, red, Muschenbroek. 1.0310 Brick, 2.000 Butter, 0.9423

C Cacao butter, 0.8916 Cachibou, gum, 1.0640 Calamine, Briffon. 3.525 La Metherie. 4.100

Calculus humanus, Campechy wood, or logwood, Muschenbroek. 1.700 Camphor, 1.240 Caoutchouc, elastic gum, or India rubber, 1.434 Caragna, resin of the Mexican tree caragna, 0.9130 Carbon of compact earth, 0.9887 Carnelian, stalagmite, 0.9335 speckled, 1.1244 veined, 2.5977 onyx, 2.6137 pale, 2.6234 pointed, 2.6227 herborisée, 2.6301 Cat's eye, Klaproth. 2.6120 grey, 2.6133 yellow, 2.625 blackish, 2.5675 Catchew, juice of an Indian tree, 2.6573 Caulitic ammonia, solution of, or fluid 1.3980 volatile alkali, 0.897 Cedar tree, American, Muschenbroek. 0.5608 Cedar, wild, Palestine, Indian, foliated, Ceylanite, Chalcedony, bluish, onyx, veined, transparent, reddish, common, Chalk, Cherry-tree, Chrysoberyl, Chrysolite of the jewellers, of Brazil, Chrysoprase, Crystal. See Rock. Crystalline lens, Cinnabar, dark red, from Deux Ponts, from Almaden, crystallized, Citron tree, Clinkstone, Cloves, volatile oil of, Cobalt, in a metallic state, fused, ore, gray, ochre, black, indurated, vitreous oxide of, Cocoa wood, Coccolite, Columbium, Copal, opaque, transparent, Madagascar, Chinese, Copper, native, from Siberia, Hungary, ore, compact vitreous, Cornish, purple, from Bannat, from Lorraine, Kirwan. 7.600 Kirwan. 7.800 Häuy. 8.5084 Gellert. 7.728 Kirwan. 4.129 Kirwan. 4.542 Kirwan. 4.956 La Metherie. 4.300 Kirwan. 4.983 Wiedemann. 5.467 Kirwan. 4.080 Briffon. 4.344 La Metherie. 4.500 Häuy. 4.865 Copper

Of Specific Gravities.

Copper ore, foliated, florid, red, Wiedemann. 3.950 azure, radiated, Wiedemann. 3.231 Briffon. 3.658 emerald, La Metherie. 2.850 Häuy. 3.300 arseniate, of, 2.549 fulphate of, saturated solution of, Watson. 1.150 temp, 42°, drawn into wire, 8.878 fuled, 7.788 Copper-sand, muriate of copper, La Metherie. 3.750 Herrgen. 4.431 Cork, Muschenbroek. 0.2400 Corundum of India, Klaproth. 3.710 Bournon. 3.875 of China, 3.981 Crofs stone, or Staurolyte, Häuy. 2.333 Heuer. 2.353 Cryolite, Karfen. 2.957 Cube iron ore, Bournon. 3.000 spar, Häuy. 2.964 Cyanite, Sauflure, jun. 3.517 Hermann. 2.622 Cyder, 1.0181 Cypres-wood, Spanish, Muschenbroek. 0.6440

D Diamond oriental, colourless, 3.5212 rose-coloured, 3.5310 orange-coloured, 3.5500 green-coloured, 3.5238 blue-coloured, 3.5254 Brazilian, 3.4444 yellow, 3.5185 Dragons blood, 1.2045

E Ebony, Indian, Muschenbroek. 1.2090 American, Muschenbroek. 1.3310 Elder tree, Muschenbroek. 0.6950 Elemi, 0.0182 Elm trunk, Muschenbroek. 0.6710 Emerald, Werner. 2.600 of Peru, Briffon. 2.7755 Häuy. 2.723 of Brasil, 3.1555 Euclaef, Häuy. 3.062 Euphorbium. 1.1244

F Fat of beef, 0.9232 veal, 0.9342 mutton, 0.9235 hogs, 0.9368 Felspar, fresh, Häuy. 2.438 Adularia, Struve. 2.500 Labrador stone, Briffon. 2.600 2.607 2.704 2.518 2.589 Filbert tree, Muschenbroek. 0.6000 Fir, male, Muschenbroek. 0.5500 female, Muschenbroek. 0.4980 Fish's eye, name of a mineral, 2.5782

Flint, olive, spotted, onyx, of Rennes, of England, variegated of Limosin, veined, Egyptian, black, Fluor, white, red, green, blue, violet, spar,

G Gadolinite, Galbanum, Galena. See Lead Glance. Galipot, a juice of the pine, Gamboge, Garnet, precious, of Bohemia, volcanic, 24 faces. of Syria, in dodecahedral crystals, common, Gas, atmospheric. See Air. Gas, azotic, pure— Barom. 29.75 Barom. 29.85 Therm. 54½ oxygenous, hydrogenous, carbonic acid, nitrous, Barom. 29.85 Therm. 54½ ammoniacal, Barom. 29.85 Therm. 54½ vapour, aqueous, sulphurous, Bar. 29.85 Ther. 54½ acid sulphurous, acid muriatic,

Blumenbach. 2.594 2.6057 2.5867 2.6044 2.6538 2.6587 2.2431 2.6122 2.5648 2.582 3.155 3.191 3.182 3.169 3.178 3.100 3.200

Häuy. 4.050 1.2120 1.0819 1.2220 4.085 4.188 Werner. 4.230 Kafner. 4.352 2.468 4.000 4.0637 Werner. 3.576 Kafner. 3.688

Lavoisier. 0.001146 0.001189 Davy. 0.001305 0.001387 0.001356 0.000999 Lavoisier. 0.000095 Dalton. 0.000123 Briffon. 0.001862 Lavoisier. 0.001845 0.001411 Kirwan. 0.001463 Briffon. 0.001302 Briffon. 0.000706 Briffon. 0.000654 Kirwan. 0.000735 Dalton. 0.000862 Sauflure. { 0.000874 Piclet. { 0.000923 Wat. { 0.000751 Kirwan. { 0.000825 { 3.131 0.002539 0.002135 Briffon. 4.000 Glance-coal,

Metheric. 1.300 Klaproth. 1.530 3.00 2.520 2.760 2.520 2.8022 2.4882 2.8348 2.7325 3.189 2.6423 2.6670 3.329 2.3959 2.5396 3.2349 2.5647 2.2694

Gold, pure, of 24 carats fine, fused, but not hammered, 19.258 the same hammered, 19.342 English standard, 22 carats fine, fused, but not hammered, 18.888 guinea of George II. 17.150 guinea of George III. 17.629 Parisian standard 22 carats, not hammered, 17.486 the same hammered, 17.589 Spanish gold coin, 17.655 Holland ducats, 19.352 trinket standard, 20 carats, not hammered, 15.709 the same hammered, 15.775 Portuguese coin, 17.9664 French money, 21 1/2 carats, fused, coined, 17.4022 French in the reign of Louis XIII. 17.5331 Granite, red Egyptian, 2.6541 gray, Egyptian, 2.7279 beautiful red, 2.7669 of Girardmor, 2.7163 violet of Gyromagny, 2.6852 red of Dauphiny, 2.6431 green, 2.6836 radiated, 2.6678 red of Semur, 2.6384 gray of Bretagne, 2.7378 yellowish, 2.6136 of Carinthia, blue, Kirwan. 2.9564 Granitelle, of Dauphiny, Muller. 3.0626 2.8465 5.723

Graphic ore, Graphite. See Plumbago. Grenatite. See Staurotide. Gum Arabic, tragacanth, 1.4523 feraphic, 1.3161 cherry tree, 1.201 Bafora, 1.4817 Acajou, 1.4346 Monbain, 1.4456 Gutte, 1.4206 ammoniac, 1.2216 Gayac, 1.2071 2.2289

Gum lac, animé d'orient, 1.1390 d'occident, 1.0284 Gunpowder in a loose heap, 1.0426 shaken, 0.836 solid, 0.932 1.745 Gypsum, opaque, compact, specimen in the Lefkean collection, 2.939 compact, 1.872 impure, 2.288 foliated, mixed with granular limestone, Kirwan. 2.725 alabaster, Ward. 1.872 semitransparent, 2.3062 fine ditto, 2.2741 opaque, 2.2642 rhomboidal, 2.3114 ditto, 10 faces, 2.3117 cuneiform, crystallised, 2.3060 flattened of France, 2.3057 of China, 2.3088 flowered, 2.3059 spathic opaque, 2.2746 semitransparent, 3.3108 granularly foliated, in the Lefkean collection, Kirwan. 2.900 mixed with marl, of a flaty form, 2.473

H Hazel, Muychenbroek. 0.606 Heavy spar, fresh, straight, lamellar, columned, not above 4.500 Heliotropium, Kirwan. 2.629 Blumenbach. 2.700 Hematites. See Ironstone. Hollow spar, Chiafolite, 2.944 Hone, razor, white, 2.8763 penetrated with water, 2.8839 razor, white and black, 3.1271 Honey, 1.4500 Honeyestone, or Mellilite, 1.586 1.666 Hornblende, common, Kirwan. 3.600 resplendent, Labrador, Kirwan. 3.830 Schiller spar, Kirwan. 3.434 2.882 schistose, Kirwan. 2.929 3.155 basaltic, Reufs. 3.150 3.220 Kirwan. 3.333 Hornstone, or Petrofilex, 2.530 ferruginous, 2.653 veined, 2.813 2.747 Hornstone, gray. See Kirwan's Mineralogy, 2.054 blackish gray, 2.744 yellowish white, 2.563 bluish, and partly yellowish gray, 2.626 dark purplish red iron shot, 2.038

Of Specific Gravities. Hornstone, greenish white, with reddish spots from Lorraine, iron shot, brownish red, outside bluish, gray inside, 2.532 Hyalite, Kirwan. 2.813 Hyacinth, Karsten. 2.110 Klaproth. 4.000 4.545 4.620 Hypocif, 1.5263

I Jade, or Nephrite, white, green, olive, 2.9592 2.9660 2.9829 from the East Indies, Kirwan. 2.977 of Switzerland, Briffon. 3.310 3.389 combined with the boracic acid and boracited calx, Mutschenbroek. 2.566 0.7700 Jasmin, Spanish, 2.6055 Jasper, veined, 2.6612 red, 2.6911 brown, 2.7101 yellow, 2.7111 violet, 2.7640 gray, 2.7354 cloudy, 2.6274 green, 2.3587 bright green, 2.6258 deep green, 2.6814 brownish green, 2.6719 blackish, 2.6277 blood coloured, 2.6339 heliotrope, 2.8160 onyx, 2.6228 flowered, red and white, 2.7500 red and yellow, 2.6839 green and yellow, 2.7323 red, green, and gray, 2.7492 red, green, and yellow, 2.5630 2.6608 universal, 1.2990 agate, 0.7690 Indigo, 1.0095 1.7228 Jet, a bituminous substance, Wollaston. 19.500 penetrated with water, Inspissated juice of liquorice, Iridium, ore of, discovered by Mr Tennant, Wollaston. 19.500 Iron, chromate of, from the department of Var, 4.0326 from the Ouralian mountains, in Siberia, Lauquier. 4.0579 Sulphate of, saturated solution, temp. 42. Wayon. 1.157 7.200 7.600 7.788 fused, but not hammered, 4.682 forged into bars, 4.830 pyrites, dodecahedral, Hatchet. 4.682 from Freyberg, Gellert. 4.789 Cornwall, Kirwan. 4.792 cubic, Briffon. 4.792 radiated, Hatchet. 4.775 sand, magnetic sand, from Virginia, 4.600

Iron ore, specular, Kirwan. 4.793 5.139 4.939 5.218 4.728 5.070 ore, specular, micaceous, Kirwan. 2.952 3.423 3.760 Ironstone, red, ochrey, compact, Wiedemann. 3.573 3.863 from Siberia, Kirwan. 3.573 3.863 Lancashire, Briffon. 3.551 3.753 compact, brown, from Bayreuth, Kirwan. 3.503 3.477 from Tyrol, cubic, Briffon. 3.503 3.477 red hematites, Kirwan. 5.005 Gellert. 4.740 brown hematites, Kirwan. 3.951 Gellert. 3.789 Wiedemann. 4.029 sparry, or calcareous, Kirwan. 3.640 3.810 Briffon. 3.672 decomposed, Kirwan. 3.300 3.600 black, compact, Wiedemann. 4.076 clay reddie, Briffon. 3.139 Blumenbach. 3.931 clay, lenticular, Kirwan. 2.673 clay, common, from Cathina at Raschau, Kirwan. 2.936 from Roscommon in Ireland, Rotheram. 3.471 Carron in Scotland, Rotheram. 3.205 clay, reniform iron ore, Wiedemann. 3.357 clay, pea ore, Molinghof. 5.207 Iron ore, lowland, from Sprottau, Kirwan. 2.944 Iferine, a mineral from the Ifer in Bohemia, 4.500 Juniper tree, Mutschenbroek. 3.563 Ivy, dry, 1.8250 Ivy gum, from the hedera terrestris, 1.2948

K Keffekil, or Meerschaum, Klaproth. 1.6000 Kinkina, Mutschenbroek. 0.7842

L Labdanum, resin, in tortis, 1.1862 2.4933 2.894 4.300 2.500 2.270 2.666 2.348 lazuli. See Azure stone. Lard, 0.9478 Lavender, volatile oil of, 0.894 Lead glance, or galena, common, Gellert. 7.290 6.565 7.786 from Derbyshire, Wayon. Lead

Lead glance, compact, crystallized, 6.886 radiated, 7.444 from the Hartz, 4.319 Kautenbach, 5.052 Kirwan, 7.587 La Methiere, 5.500 Vauquelin, 7.448 Kirwan, 6.140 Vauquelin, 6.820 Cheneux, 6.065 Bindheim, 3.920 ore, corneous, 6.745 reniform, 5.461 of black lead, 6.974 blue, brown, 6.600 from Huguelgoet, 6.909 Wiedemann, 5.770 black, white from Leadhills, 7.236 Häuy, 6.559 phosphorated from Wanlockhead, 6.560 Zschoppau, 6.270 Brilgaw, 6.941 red; or red lead spar, 5.750 yellow, molybdenated, 6.027

Lead, acetite of, 11.352 vitriol from Anglesea, 11.445 Lemon tree, 2.3953 Lenticular ore (arseniate of copper), 0.7933 Bournon, 2.882 Lepidolite, lilalite, 2.816 Häuy, 2.854 Leuzite, 2.455 Ligum vitae, 2.490 Muschenbroek, 1.3330 Limestone, compact, foliated, 1.3864 granular, 2.7200 2.710 2.837 2.700 green, 3.182 arenaceous, 2.742 white fluor, 3.156 calc. spar, 2.700 Linden wood, 0.604 Logwood, or Campeachy wood, 0.9130

M Madder root, 0.7650 Mahogany, 1.0630 Magnesia, Kirwan, 2.3300 sulphate of, saturated solution, Waton, 1.232 temp. 42°, Hatchet, 4.518 Magnetic pyrites, ironstone, 4.200 Malachite, compact, Briffon, 4.939 Briffon, 3.572 Briffon, 3.641 Muschenbroek, 3.994 Manganese, Bergman, 6.850 Hielm, 7.000 Manganese, gray ore of, striated, Briffon, 4.249 gray, foliated, Rinmann, 4.756 red from Kapnick, Kirwan, 4.181 black, Dolomieu, 3.233 penetrated with water, Briffon, 3.7076 scaly, Maple wood, Muschenbroek, 4.1165 Marble, Pyrenean, 0.7550 black Biscayan, 2.726 Brocatelle, 2.695 Caftilian, 2.650 Valencian, 2.700 Grenadian white, 2.710 Siennian, 2.705 Roman violet, 2.678 African, 2.755 Italian, violet, 2.708 Norwegian, 2.858 Siberian, 2.728 French, 2.728 Switzerland, 2.649 Egyptian, green, 2.714 yellow of Florence, 2.668 2.516 Mastic, tree, Muschenbroek, 1.0742 Medlar tree, Muschenbroek, 0.8490 Meerfschaum. See Keffekil. Melanite, or black garnet, Karsten, 3.691 Werner, 3.800 Mellilite. See Honeystone, Menachanite, Lampadius, 4.270 Gregor, 4.427 Mercurial hepatic ore, compact, Kirwan, 7.186 7.352 Gellert, 7.937 Mercury at 32° of heat, at 60°, 13.619 at 212°, 13.580 in a solid state, 40° below o Fahr, 13.375 in a fluid state, 47° above o, Biddle, 15.612 corrosive muriate of, saturated solution, temp. 42°, Waton, 1.037 natural calx of, 9.230 precipitate per fe, 10.871 red, mineralized by sulphur, native Ethiops. See also Cinabar, Hahn, 2.233 Mica, or glimmer, Briffon, 2.791 Blumenbach, 2.934 Milk, woman's, 1.0203 mare's, 1.0346 afs's, 1.0355 goat's, 1.0341 ewe's, 1.0409 cow's, 1.0324 Mineral from Cornwall, supposed to be zeolite, at 55° Fahrenheit, Gregor, 2.253

Of Specific Gravities. Mineral pitch, elastic, or asphaltum, Hatchet. { 0.905 1.233 La Metherie. 0.930 0.770 Mineral tallow, Molybdena in a metallic state, saturated with water, native, Kirwan. 7.500 Shumacher. 4.048 Briffon. 4.667 4.7383 Mountain crystal. See Rock Crystal. Mulberry tree, Spanish, Myischenbroek. 0.8970 Muricalcite, crystallized, or rhomb spar, 2.480 Myrrh, 1.3600

N Naphths, 0.8475 Nephrite. See Jade. Nickel in a metallic state, { 7.421 8.500 Bergman. 9.3333 Briffon. 6.6086 6.6481 Gellert. 7.560 Nickel, ore of, called Kupfernickel of Saxe, 6.648 Kupfernickel of Bohemia, 6.607 sulphurated, 6.620 Nickeline, a metal discovered by Richter, caft, Richter. 8.55 forged, Richter. 8.60 Nigrine, or calcareo-siliceous titanic ore, Vauquelin. 3.700 Klaproth. 4.445 Lowits. 4.673 Nitre, Myischenbroek. 1.9000 quadrangular, Myischenbroek. 2.2460 saturated solution of temperature 42° Wayton. 1.095 Novaculite, or Turkey stone. See Slate, Whet.

Oil, volatile of, tansey, Stragan, Roman camomile, sabine, fennel, fennel-feed, coriander-feed, caraway-feed, dill-feed, anife-feed, juniper-feed, cloves, cinnamon, turpentine, amber, the flowers of orange, lavender, hyfllop,

Olibanum, gum, Olive tree, copper ore, foliated, fibrous, Olivine, Opal, precious, common, semiopal, reddish, from Telkoba. nya, ligniform, or wood, Opium, Ophites. See Porphyry Hornblende. Opoponax, Orange tree, Orpiment, Orpiment, red. See Realgar.

Pear tree, Pearls, oriental, Peat, hard, Peruvian bark, Petrol, Petroflex. See Hornstone. Phosphorite, or Spargel stone, whitish, from Spain, before absorbing water, 2.8249 after absorbing water, 2.8648 greenish, from Spain, 3.098 Saxon, 3.218 Phosphorus, Pierre de volvie, Pinite, Pitch ore, or sulphurated uranite, Pitch-stone, black, yellow, red, brick red, from Misnia, leek green, inclining to olive, pearl gray, blackish,

Part I. Of Specific Gravities. { 0.9328 0.9949 0.8943 0.9294 0.9294 1.0083 0.8655 0.9249 0.0128 0.0867 0.8577 1.0363 1.0439 0.8697 0.8863 0.8798 0.8938 0.8892 1.1732 0.9270 4.281 4.281 3.225 2.114 1.958 2.015 2.144 2.540 2.600 1.3365 1.6226 0.7059 3.048 3.435 0.6610 2.683 1.329 0.7840 0.8783 2.8249 2.8648 3.098 3.218 1.714 2.320 2.980 6.378 6.330 7.500 2.0499 2.0860 2.6695 2.720 2.298 1.970 2.3191 Pitch-stone,

Of specific Gravities. Pitch-stone, olive, dark green, Britton. 2.3145 Britton. 2.3149 Pitchy iron ore, 3.956 Platina drawn into wire, 21.0417 a wedge of, sent by Admiral Gravina to Mr Kirwan, 20.663 a bar of, sent by the king of Spain to the king of Poland, 20.722 in grains purified by boiling in nitrous acid, {17.500 {18.500 native {15.601 {17.200 fused, 14.626 purified and forged, 20.336 compressed by a flatting mill, 22.069 Plum tree, Mutschenbroek. 0.7810 Plumbago, or graphite, Kirwan. {1.987 {2.267 Pomegranate tree, Mutschenbroek. 1.3549 Poplar wood, Mutschenbroek. 0.3830 white Spanish, Mutschenbroek. 0.5294 Porcelain from China, 2.3847 Seves, hard, 2.1457 tender, 2.1654 Saxony, modern, 2.4932 Limoges, 2.341 of Vienna, 2.5121 Saxony, called Petite Jaune, 2.5450 Porphyry, green, red, 2.6760 red of Dauphiny, 2.7651 red from Cordova, 2.7933 green from ditto, 2.7542 hornblende, or orphites, 2.7278 itch-stone, 2.9722 mullen, 2.412 {2.600 fand-stone, 2.728 Potash, carbonate of, 2.564 muriate of, Mutschenbroek. 1.4594 tartrite of, acidulous, Mutschenbroek. 1.8305 antimonial, 1.9000 fulphate of, 2.2460 2.2980 2.5805 Prafium, Häuy. 2.697 Prehnite of the Cape, Häuy. 2.9423 of France, Britton. 2.610 Proof spirit, according to the English excise laws, 0.916 Pumice stone, 0.9145 Pyrites, coppery, 4.9539 cubical, 4.7016 ferruginous cubic, 3.900 ditto round, 4.101 ditto of St Domingo, 3.440 magnetic. See Magnetic Pyrites. Pyrope, Klaproth. 3.718 Werner. 3.941

Q. Quartz, crystallized, brown, red, 2.6468 brittle, 2.6404 gras, 2.6459 crystallized, 2.6546 Quartz, milky, elastic, Gerhard. 2.652 Kirwan. 3.750 Mutschenbroek. 2.0240 0.7050

R. Realgar, or red orpiment, Bergman. 3.225 Britton. 3.338 Resin of guaiacum, 1.2289 of jalap, 1.2185 Rock or mountain crystal from Madagascar, 2.6530 clove brown, Karsten. 2.605 snow white from Marmeroch, Karsten. 2.888 crystal, European, pure, gelatinous, 2.6548 of Brasil, 2.6526 irifée, 2.6497 rose-coloured, 2.6701 yellow Bohemian, 2.6542 blue, 2.5818 violet, or amethyst, 2.6535 violet purple, or Carthaginian amethyst, 2.6570 pale violet, white amethyst, 2.6513 brown, 2.6534 black, 2.6536 penetrated with water, 0.5956 Ruby oriental, 1.1450 Brazilian, or occidental, 4.2833 spinell, 3.5311 3.7600 ballas, Klaproth. 3.5700 Rutile, or titanite, 3.6458 Häuy. 4.102 La Methierie. 4.246

S. Sahlite, Dandrada. 3.234 Sal gemmae, 2.143 Salt of vitriol, 1.9000 fedative, of Homberg, 1.4797 polychrest, 2.1410 de Prunelle, 2.1480 volatile, of hartshorn, 1.4760 Sandarac, 1.0920 Santal, white, Mutschenbroek. 1.0410 yellow, Mutschenbroek. 0.8090 red, Mutschenbroek. 1.1280 1.2008 Sapagenum, 3.991 Sapphire, oriental, white, 4.076 of Puys, 3.994 oriental, 3.1357 Brazilian, or occidental, Häuy. 3.994 Hatchet. 4.283 Greville. 4.083 1.2684 Sarcocolla, Britton. 2.6025 Sardonyx, pure, Britton. 2.6060 pale, Britton. 2.6215 pointed, Britton. 2.5951 veined, Britton. 2.5949 onyx, Britton. 2.5988 herborifée, Britton. 2.6284 blackish, Saffras,

Muschchenbroek. 0.4820 1.2354 1.2743 3.6800 8.7000

Silver shilling of George II. George III. French money, 10 deniers, 21 grains, fused, French money, 10 deniers, 21 grains, coined, Sinopile, coarse jaiper, Slate clay. See Argillite. common, or schifitus, common, penetrated with water, whet, or novaculite, Isabella yellow, stone, fresh polished, adhesive, new, siliceous, horn, or schistose porphyry, Smalt, or blue glaas of cobalt, Soda, sulphate of, muriate of, saturated solution, temperature 42°, tartrite of, saturated solution of, fossil, saturated solution of, temperature 42°, Sommite, or nepheline, Spar, common, heavy, brown. See Sidero-Calcite. rhomb. See Muricalcite. white sparkling, red ditto, green ditto, blue ditto, green and white do. transparent do. adamantine, or diamond, schiller. See Hornblende, Labrador. fluor, white, red, or false ruby, octahedral, yellow, or false topaz, green, or false emerald, octahedral, blue, or false sapphire, greenish blue, or false aquamarine, violet, or false amethyst, violet, purple, English, of Auvergne, in stalactites, pearled, calcareous rhomboidal, of France, prismatic, and pyramidal, pyramidal,

Of Specific Gravities.

Sassafras, 10.000 Scammony of Aleppo, 10.534 Smyrna, 10.048

Scapolite, Schifitus. See Slate, Hone, Stone. Schmelzstein, Häuy. 2.630 Schoral, black, prismatic, hexahedral, octahedral, 3.3636 emehedral, 3.2265 black, sparry, 3.0926 amorphous, or ancient basaltes, 3.3852 cruciform, 2.9225 violet of Dauphiny, 3.2861 green, 3.2956 common, 3.4529 Briffon. 3.092 Gerhard. 3.150 Kirwan. 3.212 Briffon. 3.086 tourmaline, green, Häuy. 3.362 blue, Werner. 3.155 Selenite, or broad foliated gypsum, 2.322 Serpentine, opaque, green, Italian, 2.4295 penetrated with water, 2.4729 ditto, red and black veined, 2.6273 ditto, veined, black and olive, 2.5939 semitranparent, grained, 2.5859 ditto, fibrous, 2.9997 ditto, from Dauphiny, 2.6693 opaque, spotted black and white, 2.3707 spotted black and gray, 2.2645 spotted red and yellow, 2.6885 green from Grenada, 2.6849 deep green from Grenada, 2.7907 black, from Dauphiny, or variolite, 2.9339 green from Dauphiny, 2.9883 green, 2.8960 yellow, 2.7305 violet, 2.6424 of Dauphiny, 2.7913 Siderocalcite, or brown spar, 2.837 Silver ore, sulphurated, 6.910 brittle, La Metherie. 7.200 red, Gellert. 7.208 light red, Briffon. 5.564 Briffon. 5.5886 Gellert. 5.443 footy, Vauquelin. 5.392 native, common, Gellert. 10.000 Selb. 10.333 antimonial, Häuy. 9.4406 Selb. 10.000 auriferous, Kirwan. 10.600 ore, dark red, Gellert. 5.684 Briffon. 5.5637 arseniated, ferruginous, 2.178 penetrated with water, 2.340 ore, corneous, or horn ore, Briffon. 4.7488 Gellert. 4.804 virgin, 12 deniers fine, not hammered, 10.474 12 deniers, hammered, 10.510 Paris standard, 11 deniers, 10 grains, fused, 10.175 hammered, 10.376 Chap. II.

Of Specific Gravities.

pyramidal, 2.7141 (puant gris), 2.7121 (puant noir), 2.6207 or flos ferri, 2.6747

Spargel stone, 9.9433 Spermaceti, 3.570 Spinelle, Klaproth, Wiedemann, 3.709

Spirit of wine. See Alcohol. Spodumene, Häuy, Dandrada, 3.192, 3.218

Stalactite, transparent, 2.3239 opaque, 2.4783 penetrated with water, 2.5462

Staurolite. See Crofs-stone. Staurotite, or grenatite, Häuy, 3.286 Steatites of Bareight, penetrated with water, 2.6657 indurated, 2.5834 penetrated with water, 2.6322

Steel, soft, 7.8331 hammered, 7.8404 hardened in water, 7.8103 hammered and then hardened in water, 7.8180

St John's wort, inspissated juice of, 1.5263

Strontian, Kirwan, 3.400 Klaproth, 3.644, 3.675

Stone, sand, paving, 2.4158 grinding, 2.1429 cutlers, 2.1113 Fountainbleau, glittering, 2.5616 crystallized, 2.6111 scythe, of Auvergne, mean grained, 2.5638 fine grained, 2.6090 coarse grained, 2.5686

Lorraine, 2.5298 Liege, 2.6336 mill, 2.4835 Bristol, 2.510 Burford, 2.049 Portland, 2.496 rag, 2.470 rotten, 1.981 St Cloud, 2.201 St Maur, 2.034 Notre Dame, 2.378 Clicard from Brachet; Ouchain, 2.357 rock of Chatillon, 2.274 hard paving, 2.122 Siberian blue, 2.460 touch, 2.945 prismatic basaltes, 2.415 of the quarry of Bourè, 2.722 of Chérence, 1.3864 Storax, 2.4682 Sugar, white, Mutschchenbroek, 1.1098 Sulphur, native, fused, 2.0332 Sulphuric or vitriolic acid, 1.9997 Sulphurate, triple, of lead, antimony, and copper, Hatchet, 1.841 Sylvanite, or tellurite, in a metallic state, twice fused, 5.766 Tourmaline. See Shorl. Tungsten, 6.343

Sylvan, native, ore, yellow, black, Syringa,

Jacquin, jun. 4.107 Muller. 5.723 Klaproth. 6.115 Muller. 10.678 Jacquin, jun. 6.157 Muller. 8.919 Mutschchenbroek. 1.0989

T

Tacamahaca, resin, 1.0463 Talc, black crayon, ditto German, 2.080, 2.246 yellow, 2.655 white, 2.704 of mercury, 2.7917 black, 2.9004 earthy, 2.6325 common Venetian, { 2.700 { 2.800 Tallow, 0.9419 Tantalite, 7.953 Tartar, 1.8490 Terra Japonica, 1.3980 Thumerstone, Häuy, 3.213 Gerhard, 3.300 Kirwan, 3.2956 Watson. { 7.170 { 7.291 fused and hammered, 7.291 of Malacca, fused, 7.296 fused and hammered, 7.306 of Galicia, Gellert. 7.063 of Ehrenfriedendorf in Saxony, Gellert. 7.271 Klaproth. 4.350 La Methierie. 4.785 Gellert. { 6.300 { 6.989 Brunich. 6.750 Leyster. 6.880 Briffon. 6.901 Briffon. 6.9348 Klaproth. { 5.845 { 6.970 Werner. 7.000 Brunich. 5.800 Blumenbach. 6.450 new, fused, 7.3013 fused and hammered, 7.3115 fine, fused, 7.4789 fused and hammered, 7.5194 common, 7.9200 called Claire-etoffe, 8.4869 ore, Cornish, Brunich. 5.800 Klaproth. 6.450 stone, white, Titanite. See Rutile. Topaz, oriental, 4.0106 Brazilian, 3.5395 from Saxony, 3.5640 oriental pistachio, 4.0615 Saxony white, 3.5535 Tourmaline. See Shorl. Tungsten, Leyser. 4.355 Tungsten,

Kirwan. { 5.800 { 6.028 Briffon. { 6.066 { 6.015 Klaproth. { 5.570 { 8.235 { 0.870 { 0.991 Turbeth mineral, Turpentine, spirits of, liquid, Turquoise, ivory tinged by the blue calx of copper, { 2.500 { 2.908

U Ultramarine, Deformes and Clement. 2.360 Uran, Mica, Champeauze. 3.1212 Uranite in a metallic state, Klaproth. 6.440 sulphurated. See Pitch ore. Uranitic ochre, indurated, La Metherie. 3.150 Häuy. 3.2438 Uranium, stone of, 7.500 Urine, human, { 1.015 { 1.026

V Vermeille, a kind of oriental ruby, Wiedemann. 4.2299 Veluviane, Wiedemann. 3.575 Klaproth. 3.420 of Siberia, Klaproth. { 3.365 { 3.339 Häuy. 3.407 Vine, Mutschenbroek. 1.2370 Vinegar, red, Mutschenbroek. 1.0251 white, Mutschenbroek. 1.0135 Vitriol, Dantzic, 1.715

W Walnut-tree of France, Mutschenbroek. 0.6710 Water distilled at 32° temperature, 1.0000 sea, 1.0263 of Dead sea, 1.2403 wells, 1.0017 of Bareges, 1.00037 of the Seine, filtered, 1.00015 of Spa, 1.0009 of Armeil, 1.00046 Avray, 1.00043 Seltzer, 1.0035 Wavellite, or hydragillite, Davy. 2.7000 Wax, Ourouchi, 9.8970 bees, 0.9648

Wax, white, shoemakers, 0.9686 Whey, cows, 0.807 Willow, 1.019 Mutschenbroek. 0.5850 Witherite. See Barolite. Wine of Torrins, red, 0.9930 white, 0.9876 Champagne, white, 0.9979 Pakaret, 0.9997 Xeret, 0.9924 Malmfy of Madeira, 1.0382 Burgundy, 0.9915 Juranon, 0.9932 Bourdeaux, 0.9939 Malaga, 1.0221 Conflance, 1.0819 Tokay, 1.0538 Canarry, 1.033 Port, 0.997 Wolfram, Gmelin. 5.705 Elhuyar. 6.835 Leonhardi. 7.000 Hatchet. 6.955 Häuy. 7.333 2.3507 2.045 2.675

Wolf's eye (name of a mineral), Y Woodstone,

Yew tree, Dutch, Mutschenbroek. 0.7880 Spanish, Mutschenbroek. 0.8070 Yttertantalite, Eckeberg. 5.130

Z Zeolite from Edelfors, red, scintillant, 2.4868 white scintillant, 2.0739 compact, 2.1344 radiated, Häuy. 2.083 cubic, Häuy. 2.716 siliceous, 2.515 Zinc, pure and compressed, 7.1908 in its usual state, 6.862 formed by sublimation and full of cavities, Kirwan. 5.918 sulphate of, Mutschenbroek. 1.0000 saturated solution of, temp. 42°, Watson. 1.386 Zircon, or jargon, Klaproth. 4.615 Karsten. 4.666 Wiedemann. 4.700

CHAP. III. On Capillary Attraction, and the Cohesion of Fluids.

III. 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 its phenomena are named capillary tubes. These appellations derive their origin from the Latin word capillus, signifying a hair, either either because the bores of these tubes have the fineness of a hair, or because that substance is itself supposed to be of a tubular structure.

112. 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 seems to resist 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 is exhibited in fig. 2, where AB is the metallic plate, C, D the drops of water, and m, n the two canals.

113. Upon these principles many attempts have been made to account for the elevation of water in capillary tubes; but all 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 E be a drop of water laid upon a clean glass surface AB. Every particle of the glass immediately below the drop E, 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 this attractive force, therefore, tending to press the drop to the glass will be an enlargement of its size, and the water will occupy the space FG; 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. 2.) be a section of the plate of glass AB (fig. 3.) 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 o p. The gravity of the drops B and D being thus diminished, the film of water at m and n which was 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 o p), 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 the altitudes 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, their diameters, 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 D^a A = d^a D, and DA = \frac{d^a}{D^a} D, or DA = d a, 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 0.01, or \frac{1}{100} of an inch, the water has been found to reach the altitude of 5.3 inches; hence the constant quantity 5.3 \times 0.01 = 0.053 may fitly represent the attraction of glass for water. According to the experiments of Mutchenbroek, the constant quantity is 0.039; according to Weitbrecht 0.0428; according to Monge 0.042, and according to Atwood 0.053. When a glass tube was immersed in melted lead, Gellert found the depression multiplied by the bore to be 0.054.

114. Having thus attempted to explain the causes of capillary action, we shall now proceed to consider of some of its most interesting phenomena. In fig. 4. 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 top, whatever be the length of the tube or the diameter of 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. These two facts seem to countenance the opinion of Dr Jurin* pothes. and other philosophers, that the water is elevated in * Phil. the tube by the attraction of the annulus, or ring of glass, immediately above the cylinder of water. This No. 363. art. 2. hypothesis is sufficiently plausible; but supposing it to be true, the ring of glass immediately below the surface of the cylinder of fluid should produce an equal and opposite effect, and therefore the water instead of rising should be stationary, being influenced by two forces of an equal and opposite kind.

115. If a tube D composed of two cylindrical tubes Phenomena of different bores be immersed in water with the widest of capillary 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 Capillary raises much less water than D. But if we admit the supposition in art. 113, 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.

Phenomena of capillary attraction.

Fig. 4.

116. When a vessel F v w is plunged in water, and the lower part t u x w filled by suction till the fluid enter the part F t, 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 F t. In this experiment the portions of water t u x and x w on each side of the column F v are supported by the pressure of the atmosphere on the surface of the water in the vessel MM; for if this vessel be placed in the exhausted receiver of an air-pump, these portions of water will not be sustained. Dr Jurin, indeed, maintains that these portions will retain their position in vacuo, but in his time the exhausting power of the air-pump was not sufficiently great to determine a point of so great nicety. The column t u x, which is not sustained by atmospherical pressure, is kept in its position by the attraction of the water immediately around and above it, and the column F t u is supported by the attraction of the glass surface with which it is in contact. According to Dr Jurin's hypothesis, the column t u x is supported by the ring of glass immediately above r, which is a very unlikely supposition.

117. The preceding experiment completely overturns the hypothesis of Dr Hamilton, afterwards revived by Dr Matthew Young. These philosophers maintained that the fluid was sustained in the tube by the lower ring of glass contiguous to the bottom of the tube, that this ring raised 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. But if the elevation of the fluid were produced in this way, the quantity supported would be regulated by the form and magnitude of the orifice at the bottom of the tube; whereas it is evident from every experiment, that the cylinder of fluid sustained in capillary tubes has no reference whatever to the form of the lower annulus, but depends solely upon the diameter of the tube immediately above the elevated column of water.

118. If the experiments which we have now explained be performed in the exhausted receiver of an air-pump, the water will rise to the same height as when they are performed in air. We may therefore conclude, that the ascent of the water is not occasioned, as some have imagined, by the pressure of the atmosphere acting more freely upon the surface of the water in the vessel than upon the column of fluid in the capillary tube.

119. It appears from the following table constructed by Mr B. Martin, that different fluids rise to very different heights in capillary tubes, and that spirituous liquors whose specific gravity is less than that of water, are not raised to the same altitude. Mr Martin's experiments were made with a tube about \( \frac{1}{4} \) of an inch in diameter. He found that when capillary tubes charged with different fluids were suspended in the sun for months together, the enclosed fluid was not in the least degree diminished by evaporation.

Names of the Fluids.

<table> <tr> <th>Names of the Fluids.</th> <th>Constant Number</th> <th>Altit.</th> </tr> <tr> <td>Common spring water</td> <td>.048</td> <td>Inches 1.2</td> </tr> <tr> <td>Spirit of urine</td> <td>.044</td> <td>1.1</td> </tr> <tr> <td>Tincture of galls</td> <td>.044</td> <td>1.1</td> </tr> <tr> <td>Recent urine</td> <td>.044</td> <td>1.1</td> </tr> <tr> <td>Spirit of salt</td> <td>.036</td> <td>0.9</td> </tr> <tr> <td>Ol. tart. per deliq.</td> <td>.036</td> <td>0.9</td> </tr> <tr> <td>Vinegar</td> <td>.038</td> <td>0.95</td> </tr> <tr> <td>Small beer</td> <td>.036</td> <td>0.9</td> </tr> <tr> <td>Strong spirit of nitre</td> <td>.034</td> <td>0.85</td> </tr> <tr> <td>Spirit of hartthorn</td> <td>.034</td> <td>0.85</td> </tr> <tr> <td>Cream</td> <td>.032</td> <td>0.8</td> </tr> <tr> <td>Skimmed milk</td> <td>.032</td> <td>0.8</td> </tr> <tr> <td>Aquafortis</td> <td>.030</td> <td>0.75</td> </tr> <tr> <td>Red wine</td> <td>.030</td> <td>0.75</td> </tr> <tr> <td>White wine</td> <td>.030</td> <td>0.75</td> </tr> <tr> <td>Ale</td> <td>.030</td> <td>0.75</td> </tr> <tr> <td>Ol. ful. per campanam</td> <td>.026</td> <td>0.65</td> </tr> <tr> <td>Oil of vitriol</td> <td>.026</td> <td>0.65</td> </tr> <tr> <td>Sweet oil</td> <td>.024</td> <td>0.6</td> </tr> <tr> <td>Oil of turpentine</td> <td>.022</td> <td>0.55</td> </tr> <tr> <td>Geneva</td> <td>.020</td> <td>0.5</td> </tr> <tr> <td>Rum</td> <td>.020</td> <td>0.5</td> </tr> <tr> <td>Brandy</td> <td>.020</td> <td>0.5</td> </tr> <tr> <td>White hard varnish</td> <td>.020</td> <td>0.5</td> </tr> <tr> <td>Spirit of wine</td> <td>.018</td> <td>0.45</td> </tr> <tr> <td>Tincture of mars</td> <td>.018</td> <td>0.45</td> </tr> </table>

120. To the preceding table as given by Mr Martin we have added the constant number for each fluid, or the product of the altitude of the liquid, and the diameter of the tube (art. 113.). By this number, therefore, we can find the altitude to which any of the preceding fluids will rise in a tube of a given bore, or the diameter of the bore when the altitude of the fluid is known; for since the constant number C=DA (art. 113.) we shall have

\[ D = \frac{C}{A} \quad \text{and} \quad A = \frac{C}{D}. \]

Since the constant number, however, as deduced from the experiments of Martin, may not be perfectly correct, it would be improper to derive from it the diameter of the capillary bore when great accuracy is necessary. The following method, therefore, may be adopted as the most correct that can be given. Put into the capillary tube a quantity of mercury, whose weight in troy grains is W, and let the length L of the tube which it occupies be accurately ascertained; then if the mercury be pure and at the temperature of 60° of Fahrenheit, the diameter of the capillary tube

\[ D = \sqrt{\frac{W}{L}} \times 0.019241, \]

the specific gravity of mercury being 13.380. The weight of a cubic inch of mercury being 3438 grains, and the solid content of the mercurial column being D^2L×0.7854, we shall have \( 1 : 3438 = D^2L \times 0.7854 : W \). Hence (GEOMETRY, Sect IV. Theor. VIII.) \( D^2L \times 0.7854 \times 3438 = W \), and dividing we have \( D^2 = \frac{L \times 0.7854 \times 3438}{W} \)

or \( D = \sqrt{\frac{W}{L \times 0.7854 \times 3438}} \) or \( D = \sqrt{\frac{W}{L}} \times 0.019241 \).

If the whole tube be filled with mercury, and if W be the difference in troy grains between its weight when empty, empty, and when filled with mercury, the same theorem will serve for ascertaining the diameter of the tube. Should the temperature of the mercury happen to be 32° of Fahrenheit, its specific gravity will be 13.619, which will alter a very little the constant multiplier 0.019241.

121. 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. Mr Leflic 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 syphon also which discharged cold water only by drops, yielded warm water in an invariable stream.

122. 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 AB E F and C D E F be two pieces of plate glass with smooth and clean surfaces, having their sides EF joined together with wax, and their sides AB, CD kept 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 D q o m E, which represents the surface of the elevated water. By measuring the ordinates m, o, p, &c. of this curve, and also its abscisse F n, F p, &c. Mr Hawkingbee found it to be the common Apollonian hyperbola, having for its asymptotes the surface DF of the fluid, and EF the common intersection of the two 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 D q o m E from D towards E, the water ought to rise higher at o than at q, still higher at m, and highest of all at E, where the distance between the plates is a minimum. To illustrate this more clearly, let ABEF and CDEF be the same plates of glass, (inclined at a greater angle for the sake of distinctness) and let E m g D, and E o s B 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 m o, or its equal n p, and the distance at q is q s or r t; and since m n is the altitude of the water at m, and q r its altitude at q, we have m n : q r = n p : r t; but (Geometry, Sect. IV. Theor. XVII.) F n : Fr = n p : r t; therefore m n : q r = F n : Fr; that is, the altitudes of the fluid at the points m, q, which are equal to the abscisse F n, Fr (fig. 5.) are proportional to the ordinates q r, m n, equal to F n, Fr, in fig. 5. But in the Apollonian hyperbola the ordinates are inversely proportional to their respective abscisse, therefore the curve D q o m E is the common hyperbola.—As the plates are infinitely near each other at the apex E, the water will evidently rise to that point, whatever be the height of the plates.

123. The phenomena which we have been endeavouring to explain, are all referable 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. 7.) be a vessel full of mercury, Fig. 7. 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.

124. 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 then for 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 Caufe of film of fluid, this drop, if very small, will be spherical, the depression of mercury in contact with it will be very small; 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 upper poles, like AB in fig. 8. The drop, however, will Fig. 8. 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 o A m, p B n, 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 m, n, o, p, 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. 9. Fig. 9. If the drop AB fig. 8. be now supposed mercury instead of water, it will also, by the gravity of its parts, assume the form of an oblate spheroid; but when the pieces of glass o A m, p B n 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 of AB in fig. 10. The small spaces o, p being filled Fig. 10. by the prelude of the superincumbent fluid, while the spaces m, n, fill 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 Capillary 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. 5, 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 D O E, having for its asymptotes the common intersection of the planes and the surface of mercury in the vessel.

The depression of mercury in glass tubes is evidently owing to the greater attraction that subsists between the particles of mercury, than between the particles of mercury and those of 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.

Capillary attraction does not seem to act at any perceptible distance.

Newton, Clairaut, and other geometers, have maintained, that the action of the capillary tube is sensible at a small distance, and that it is extended to the particles of fluid in the axis of the tube. Laplace and other philosophers who have lately attended to this subject, suppose capillary attraction to be like the refractive force, and all the chemical affinities, which are not sensible except at imperceptible distances; and it must be allowed that this opinion is consistent with many of the phenomena. It has been often observed that water rises to the same height in glass tubes, of the same bore, whether they be very thin or very thick. The zones of the glass tube therefore, which are at a small distance from the interior surface, do not contribute to the ascent of the water, though in each of these zones, taken separately, the water would rise above its level. When the interior surface of a capillary tube is lined with a very thin coating of an unctuous substance, the water will no longer ascend. Now if the attraction of the glass tube were similar to the attraction of gravity, of electricity, or magnetism, it ought to act through bodies of all kinds, and, notwithstanding the thin coating of grease, should elevate the fluid in which it is immersed. But as the intervention of an attenuated film of grease destroys capillary action, there is reason to conclude, that it does not extend to sensible distances. The same conclusion is deducible from the fact in the preceding paragraph.

Opinion of Laplace.

127. From these facts Laplace concludes, that the attraction of capillary tubes has not any influence on the elevation or depression of the fluids which they contain, except by determining the inclination of the first planes of the surface of the interior fluid, which are extremely near the sides of the tube. He supposes that when the attraction of the tube upon the fluid exceeds the attraction of the fluid upon itself, the fluid will in that case attach itself to the tube, and form an interior tube, which alone will raise the fluid.

128. 'It is interesting,' says Laplace, to ascertain the radius of curvature of the surface of water included in capillary tubes of glass. This may be known by a curious experiment, which shews at the same time the effects of the concavity and convexity of surfaces. It consists in plunging in water, to a known depth, a capillary tube of which the diameter is likewise known. The lower extremity of the tube is then to be closed with the finger, and the tube being taken out of the water, its external surface must be gently wiped. Upon withdrawing the finger in this last situation, the water is seen to subside in the tube and form a drop at its lower base; but the height of the column is always greater than the elevation of the water in the tube above the level in the common experiment of plunging it in water. This excess in the height is owing to the action of the drop upon the column on account of its convexity; and it is observable that the increase in the elevation of the water is more considerable, the smaller the diameter of the drop beneath. The length of the fluid column which came out by subidence to form the drop, determines its mass; and as its surface is spherical as well as that of the interior fluid, if we know the height of the fluid above the summit of the drop, and the distance of this summit from the plane of the interior bore of the tube, it will be easy to deduce the radii of these two surfaces. Some experiments lead me to conclude that the surface of the interior fluid approaches very nearly to the figure of an hemisphere.'

129. 'The theory which I have adopted, observes the fame philosopher, likewise gives the explanation and measure of a singular phenomenon presented by experiment. Whether the fluid be elevated or depressed between two vertical planes, parallel to each other, and plunged in the fluid at their lower extremities, the planes tend to come together. Analysis shews us, that if the fluid be raised between them, each plane will undergo from without inwards a pressure equal to that of a column of the same fluid, of which the height would be half the sum of the elevations above the level of the points of contact of the interior and exterior surfaces of the fluid with the plane, and of which the base should be the parts of the plane comprised between the two horizontal lines drawn through those points. If the fluid be depressed between the planes, each of them will in like manner undergo from without inwards, a pressure equal to that of a column of the same fluid, of which the height would be half the sum of the depressions below the level of the points of contact of the interior and exterior surfaces of the fluid with the plane, and of which the base should be the part of the plane comprised between the two horizontal lines drawn through those points.'

130. 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 force 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, 1\frac{1}{2} French inches in diameter, required a weight of 91 French grains to raise it.

Attraction, which is only 37 English grains for each square inch, &c.

At 44 1/2° of Fahrenheit the force was 1 1/2 greater, or 39 1/2 grains, the difference being 1 1/27 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 1 8/50 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 1 1/200. According to the experiments of Dutour, the force necessary to elevate the solid, or the quantity of water raised, is equal to 44.1 grains for every square inch.

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; 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 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.

A number of experiments on the adhesion of fluids have been lately made by Count Rumford, which authorise him to conclude, that on account of the mutual adhesion of the particles of fluid, a pellicle or film is formed at the superior and inferior surfaces of water, and that the force of the film to resist the descent of bodies specifically heavier than the fluid increases with the viscosity of the water. He poured a stratum of sulphuric ether upon a quantity of water, and introduced a variety of bodies specifically heavier than water into this compound fluid. A sewing needle, granulated tin, and small globules of mercury, descended through the ether, but floated upon the surface of the water. When the eye was placed below the level of the aqueous surface, the floating body, which was a spherule of mercury, seemed suspended in a kind of bag a little below the surface. When a larger spherule of mercury was employed, about the 40th or 50th of an inch in diameter, it broke the pellicle and descended to the bottom. The same results were obtained by using essential oil of turpentine or oil of olives instead of either. When a stratum of alcohol was incumbent upon the water, a quantity of very fine powder of tin thrown upon its surface, descended to the very bottom, without seeming to have met with any resistance from the film at the surface of the water. This unexpected result Count Rumford endeavours to explain by supposing that the aqueous films was destroyed by the chemical action of the alcohol. In order to ascertain with greater accuracy the existence of a pellicle at the surface of the water, Count Rumford employed a cylindrical glass vessel 10 inches high and 1 1/2 inch in diameter, and filled it with water and ether as before. A number of small bodies thrown into the vessel descended through the ether and floated on the surface of the water. When the whole was perfectly tranquil, he turned the cylinder three or four times round with considerable rapidity in a vertical position. The floating bodies turned round along with the glass, and stopped when it was stopped; but the liquid water below the surface did not at first begin to turn along with the glass; and its motion of rotation did not cease with the motion of the vessel. From this Count Rumford concludes that there was a real pellicle at the surface of the water, and that this pellicle was strongly attached to the sides of the glass, so as to move along with it. When this pellicle was touched by the point of a needle, all the small bodies upon its surface trembled at the same time. The apparatus was allowed to stand till the ether had entirely evaporated, and when the pellicle was examined with a magnifier, it was in the same state as formerly; and the floating bodies had the same relative positions.

In order to shew that a pellicle was formed at the inferior surface of water, Count Rumford poured water upon mercury, and upon that a stratum of ether. He threw into the vessel a spherule of mercury about one-third of a line in diameter, which being too heavy to be supported by the pellicle at the superior surface of the water, broke it, and descending through that fluid, was stopped at its inferior surface. When this spherule was moved, and even compressed with a feather, it still preserved its spherical form, and refused to mix with the mass of mercury. When the viscosity of the water was increased by the infusion of gum-arabic, much larger spherules were supported by the pellicle. From the very rapid evaporation of ether, and its inability to support the lightest particles of a solid upon its surface, Count Rumford very justly concludes, that the mutual adhesion of its particles is very small.

Those who wish to extend their inquiries concerning the cohesion of fluids, may consult an ingenious paper on Capillary Action by Professor Leslie, in the Phil. Mag. for 1802; Dr Thomas Young's Essay on the Cohesion of Fluids, in the Phil. Trans. 1805; an Abstract of a Memoir of Laplace, in Nicholson's Journal, No. 57.; and an Account of Rumford's Experiments in the same Journal, No. 60, 61, and 62. PART II. HYDRAULICS.

Definition. 136. 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’eaux 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.

CHAP. I. Theory of Fluids issuing from Orifices in Reservoirs, either in a Lateral or a Vertical direction.

Preliminary observations. 137. 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. 1.; but when they have reached within three or four inches of the orifice m n, 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 m n 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 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 o p than the width of the orifice m n.

138. 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 o p is also manifest from observation. It was first discovered by Sir Isaac Newton, and denominated the vena contracta. The greatest contraction takes place at a point o whose distance from the orifice is equal to half its diameter, so that o m = \( \frac{m n}{2} \); and the breadth of the vein or column of fluid at o is to the width of the orifice as 5 to 8 according to Boffiat, or as 5.197 to 8 according to the experiments of Michellotti, the orifice being perforated in a thin plate. But when the water is made to issue through a short cylindrical tube, the same contraction, though not obvious to the eye, is so considerable, that the diameter of the contracted vein is to that of the orifice as 6.5 to 8. If A therefore be the real size of the orifice in a thin plate, its corrected size, or the breadth of the contracted vein, will be \( \frac{5.197 \times A}{8} \), and when a cylindrical tube is employed it will be \( \frac{13 \times A}{16} \). In the first case the height of the water in the reservoir must be reckoned from the surface of the fluid to the point o, where the vein ceases to contract; and when a cylindrical tube is employed, it must be reckoned from the same surface to the exterior aperture of the tube.

139. Suppose the fluid ABCD divided into an infinite number of equal strata or laminae by the horizontal surfaces MN, gh infinitely near each other; and let m n o p be a small column of fluid which issues from the orifice in the same time that the surface MN descends to gh. The column m n o p is evidently equal to the that of the lamina MN gh, for the quantity of fluid which is discharged during the time that MN descends to gh, is evidently MN gh; and to the quantity discharged in that time, the column m n o p was equal by hypothesis. Let A be the area of the base MN, and B the area of the base mn; let x be the height of a column equal to MN gh, and having A for its base, and let y be the height of the column m n o p. Then, since the column m n o p is equal to the lamina MN gh, we shall have A x = B y, and (GEOMETRY, Sect. IV. Theor. IX.) x : y = A : B; but as the surface MN descends to gh in the same time that m n o p descends to o p, x will represent the mean velocity of the lamina MN gh, and y the mean velocity of the column m n o p. The preceding analogy, therefore, informs us, that the mean velocity of any lamina is to the velocity of the fluid issuing from the orifice reciprocally as the area of the orifice is to the area of the base of the lamina MN gh. Hence it follows, that, if the area of the orifice is infinitely small, with regard to the area of the base of the lamina into which the fluid is supposed to be divided, the mean velocity of the fluid at the orifice 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.

140. 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 (32.). In this way they have represented the motion of fluids in general formulae; but these formulae are so complicated from the

Motion of very nature of the theory, and the calculations are fo intricate, and sometimes impracticable from their length, that they can afford no assistance to the practical engineer.

DEFINITION.

141. If the water issues at \( m n \) 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 \( V \).

PROP. I.

142. The velocity of a fluid issuing from an infinitely small orifice in the bottom or side of a vessel, is equal to that which is due to the height of the surface of the fluid above that orifice, the vessel being supposed constantly full.

Let \( AB \) be the vessel containing the fluid, its velocity when issuing from the aperture \( m n \) will be that which is due to the height \( D m \), or equal to that which a heavy body would acquire by falling through that height. Because the orifice \( m n \) is infinitely small, the velocity of the laminae into which the fluid may be supposed to be divided, will also be infinitely small (art. 138.). But since all the fluid particles, by virtue of their gravity, have a tendency to descend with the same velocity; and since the different laminae of the fluid lose this velocity, the column \( m n s t \) must be preflled by the superincumbent column \( D m n \); and calling \( S \) the specific gravity of the fluid, the moving force which pushes out the column \( m n s t \) will be \( S \times D m \times m n \) (art. 42.). Now let us suppose, that, when this moving force is pushing out the column \( m n s t \), the absolute weight of the column \( m n o p \), which may be represented by \( S \times m n \times n p \), causes itself to fall through the height \( n p \). Thus, if \( V \), \( U \) be the velocities impressed upon the columns \( m n s t \), and \( m n o p \), by the moving forces \( S \times D m \times m n \), and \( S \times m n \times n p \); these moving forces must be proportional to their effects, or to the quantities of motion which they produce, that is, to \( V \times m n s t \) and \( U \times m n o p \), because the quantity of motion is equal to the velocity and mass conjointly; hence we shall have \( S \times D m \times m n : S \times m n \times n p = V : U \times m n s t : U \times m n o p \). But since the volumes \( m n s t \), \( m n o p \) are to one another as their heights \( m o \), \( o r \), and as their heights are run through in equal times, and consequently represent the velocity of their motion, \( m n s t \) may be represented by \( V \times m n \) and \( m n o p \) by \( U \times m n \); therefore we shall have \( S \times D m \times m n : S \times m n \times n p = V \times V \times m n : U \times U \times m n \), and dividing by \( m n \), \( S \times D m : n p = V^2 : U^2 \). Now let \( v \) be the velocity due to the height \( D m \), then (see MECHANICS) \( n p : U^2 = D m : v^2 \); but since \( S \times D m : S \times n p = V^2 : U^2 \); then by (Euclid V. 15.), and by permutation \( D m : V^2 = n p : U^2 \), therefore by substitution (Euclid V. 11.) \( D m : V^2 = D m : v^2 \), and (Euclid V. 9.) \( V^2 = v^2 \) or \( V = v \). But \( V \) is the velocity with which the fluid issues from the orifice \( m n \), and \( v \) is the velocity due to the height \( D m \); therefore, since the velocities are equal, the proposition is demonstrated.

143. Cor. 1. If the vessel \( AB \) empties itself by the small orifice \( m n \), so that the surface of the fluid takes successively the positions DP, QR, ST, the velocities with which the water will issue when the surfaces have these positions will be those due to the heights \( E n, F n, G n \), for in these different positions the moving forces are the columns \( E m n, F m n, G m n \).

144. Cor. 2. Since the velocities of the issuing fluid when its surface is at \( E, F, G \), are those due to the heights \( E n, F n, G n \), it follows from the properties of falling bodies (see MECHANICS), that if these velocities were continued uniformly, the fluid would run through spaces equal to \( 2 E n, 2 F n, 2 G n \) respectively, in the same time that a heavy body would fall through \( E n, F n, G n \); respectively.

145. Cor. 3. 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. This corollary holds also in the case mentioned in Cor. 1.

146. Cor. 4. 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; for (see MECHANICS), this is true of falling bodies in general, and must therefore be true in the case of water: 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.

147. Cor. 5. As the velocities of falling bodies are as the square roots of the heights through which they fall (see MECHANICS), the velocity \( V \) of the effluent water when the surface is at \( E \), will be to its velocity \( v \) when the surface is at \( G \), as \( \sqrt{E n} : \sqrt{G n} \), (Cor. 1.) that is, 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 fluid 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, notwithstanding Belidor (Architect. Hydraul. tom. i. p. 187.) has maintained the contrary; for though a column of mercury \( D m n \) presses with 14 times the force of a similar column of water, yet the column \( m n o p \) of Fig. 2, mercury which is pushed out is also 14 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.

148. Cor. 6. 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 (art. 138.) 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, \( V : V' = v : v' \); but, (Cor. 1.) \( v : v' = \sqrt{h} : \sqrt{h'} \), \( h \) 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.

149. Cor. 7. 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 emp-

Motion of Fluids, &c.

For (Cor. 2. and 6.) the space through which the surface of the fluid at D would descend if its velocity continued uniform being 2 Dm, double of Dm 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.

SCHOLIUM.

150. 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 vena 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 vena contracta, Sir Isaac found that the velocity at the vena 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 vena contracta as (H) 1 : \( \sqrt{2} \), and in 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 reconciled. The velocity found by the preceding proposition is evidently the vertical velocity of the filaments at E, which being immediately above the centre of the aperture mn are not diverted from their course, and have therefore their vertical equal to their absolute velocity. But the vertical velocity of the particles between C and E, and E and D, 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 particles at E, that is, than the velocity found by the above proposition, or that due to the height Dm. 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 our proposition gives the velocity of the particles at D, or the velocity at the vena contracta.

(h) When a fluid runs through a conical tube kept continually full, the velocities of the fluid in different sections will be inversely as the area of the sections. For as the same quantity of fluid runs through every section in the same time, it is evident that the velocity must be greater in a smaller section, and as much greater as the section is smaller, otherwise the same quantity of water would not pass through each section in the same time. Now the area of the vena contracta is to the area of the orifice, as 1 : \( \sqrt{2} \), therefore the velocity at the vena contracta must be to the velocity at the orifice as \( \sqrt{2} : 1 \).

Part II.

Prop. II.

151. To find the quantity of water discharged from a very small orifice in the side or bottom of a reservoir, the time of discharge, and the altitude of the fluid, the vessel being kept constantly full, and any two of these quantities being given.

Let A be the area of the orifice mn; W the quantity of water discharged in the time T; H the constant height Dm of the water in the vessel, and let 16.087 feet be the height through which a heavy body descends in a second of time. Now, as the times of description are proportional to the square roots of the heights described, the time in which a heavy body will fall through the height H, will be found from the following analogy, \( \sqrt{16.087} : \sqrt{H} = 1 : \frac{\sqrt{H}}{16.087} \), the time required. But as the velocity at the orifice is uniform, a column of fluid whose base is mn and altitude 2H (Prop. I. Cor. 2.) will issue in the time 16.087 \( \sqrt{H} \), or since A is the area of the orifice mn, A \( \times \) 2H or 2HA will represent the column of fluid discharged in that time. Now since the quantities of fluid discharged in different times must be as the times of discharge, the velocity at the orifice being always the same, we shall have \( \frac{\sqrt{H}}{16.087} : T = 2HA : W \), and (GEOMETRY, Sect. IV. Theor. VIII.)

\[ W \sqrt{H} = 2HAT \times 16.087 \]

\[ \frac{W}{16.087} = 2HAT \text{ or } W = \frac{2HAT \times 16.087}{\sqrt{H}} \]

and since

\[ \frac{H}{\sqrt{H}} = \sqrt{H} \]

we shall have

\[ W = 2AT \sqrt{H} \times 16.087 \]

an equation from which we deduce the following formulae, which determine the quantity of water discharged, the time of discharge, the altitude of the fluid, and the area of the orifice, any three of these four quantities being given:

\[ W = 2AT \sqrt{H} \times 16.087 \quad A = \frac{W}{2T \sqrt{H} \times 16.087} \]

\[ H = \frac{W^2}{4A^2T^2 \times 16.087} \quad T = \frac{W}{2A \sqrt{H} \times 16.087} \]

152. 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. 3.), the depths of different parts of the orifice below the surface of the fluid are very different, and consequently the preceding formulae will not give very accurate results.

Motion of fluids. 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 D a, the water at b, with a velocity due to the height E b, and the water at c, with a velocity due to the height Fc. 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.

PROP. III.

153. To find the quantity of water discharged by a rectangular orifice in the side of a vessel kept constantly full.

Let ABD be the vessel with the rectangular orifice CCLXVII. GL, and let AB be the surface of the fluid. Draw the lines MNOP, m n o p infinitely near each other, and from any point D draw the perpendicular DC meeting the surface of the fluid in C. Then regarding the infinitely small rectangle MO mo as an orifice whose depth below the surface of the fluid is H, we shall have by the first of the preceding formulae, the quantity of water discharged in the time T, or W = \( \frac{T}{\sqrt{16.087}} \times \sqrt{CN} \times 2MO \times NN_n \), CN being equal to H and MO × N n to the area A. As the preceding formula represents the quantity of fluid discharged by each elementary rectangular orifice, into which the whole orifice GL is supposed to be divided, we must find the sum of all the quantities discharged in the time T, in order to have the total quantity afforded by the finite orifice in the same time. Upon DC as the principal axis, describe the parabola CHE, having its parameter P equal to 4DC. Continue FG and DK to H and E. The area NP p n may be expressed by NP × N n. But (Conic Sections, Part I. Prop. X.) NP² = CN × P (P being the parameter of the parabola) therefore NP = \( \sqrt{CN \times P} \), and multiplying by N n we have NP × N n = N n / \( \sqrt{CN \times P} \), which expresses the area NP p n. Now this expression of the elementary area being multiplied by the constant quantity \( \frac{T}{\sqrt{16.087}} \times \frac{MO}{\sqrt{\frac{1}{4}P}} \) gives for a product \( \frac{T}{\sqrt{16.087}} \times \sqrt{CN} \times 2MO \times N n \), for \( \sqrt{\frac{1}{4}P} = \frac{1}{2} \sqrt{P} \) and \( \frac{MO \times \sqrt{P}}{\frac{1}{4}P} = 2MO \). But that product is the very same formula which expresses the quantity of water discharged in the time T by the orifice MO o m. Therefore since the elementary area MP p m multiplied by the constant quantity \( \frac{T}{\sqrt{16.087}} \times \frac{MO}{\sqrt{\frac{1}{4}P}} \) gives the quantity of water discharged by the orifice MO o m in a given time, and since the same may be proved of every other orifice of the same kind into which the whole orifice is supposed divided, we may conclude that the quantity of water discharged by the whole orifice GL will be found by multiplying the parabolic area FHED by the same constant quantity \( \frac{T}{\sqrt{16.087}} \times \frac{MO}{\sqrt{\frac{1}{4}P}} \). Now the area FHED is equal to the difference between the areas CDE and CFH. But (Conic Sections, Part I. Prop. X.) the area CDE = \( \frac{1}{2} CD \times DE \); and since P = 4CD, and (Conic Sections, Part I. Prop. X.) \( DE^2 = CD \times P \) we have \( DE^2 = CD \times 4CD = 4CD^2 \), that is \( DE = 2CD \), then by substituting this value of DE in the expression of the area CDE, we have \( CDE = \frac{1}{2} CD^2 \). The area CFH = \( \frac{1}{2} CF \times FH \), consequently the area FHED = \( \frac{1}{2} CD^2 - \frac{1}{2} CF \times FH \), which multiplied by the constant quantity, gives for the quantity of water discharged, (\( \frac{1}{4} P \) being substituted instead of its equal \( \frac{1}{4} CD^2 \))

\[ W = \frac{T}{\sqrt{16.087}} \times \frac{MO \times \frac{1}{4}P^2 - \frac{1}{2}CF \times FH}{\sqrt{\frac{1}{4}P}} \]

But by the property of the parabola FH² = CF × P and FH = \( \sqrt{CF \times P} \), therefore substituting this value of FH in the preceding formula, and also \( \sqrt{\frac{1}{4}P} \) for its equal \( \sqrt{\frac{1}{4}P} \), we have

\[ W = \frac{T}{\sqrt{16.087}} \times \frac{MO \times \frac{1}{4}P^2 - \frac{1}{2}CF \times \sqrt{CF \times P}}{\frac{1}{2}\sqrt{P}} \]

and dividing by \( \frac{1}{2}\sqrt{P} \) gives us

\[ W = \frac{T}{\sqrt{16.087}} \times MO \times \frac{\frac{1}{4}P^2 - \frac{1}{2}CF \times \sqrt{CF}}{\sqrt{P}} \]

hence

\[ T = \frac{W}{\sqrt{16.087} \times MO \times \frac{1}{4}P \sqrt{P} - \frac{1}{2}CF \times \sqrt{CF}} \]

\[ MO = \frac{W}{\frac{T}{\sqrt{16.087}} \times \frac{1}{4}P \sqrt{P} - \frac{1}{2}CF \times \sqrt{CF}} \]

\[ P = \frac{9W}{4T \sqrt{16.087}} + 3CF \sqrt{CF} \]

and since P = 4CD

\[ CD = \frac{9W}{16T \sqrt{16.087}} + 12CF \times \sqrt{CF} \]

\[ CF = \frac{9W}{16T \sqrt{16.087}} + \frac{1}{8}P \sqrt{P} \]

In these formulae W represents the quantity of water discharged, T the time of discharge, MO the horizontal width of the rectangular orifice, P the parameter of the parabola = 4CD, CD the depth of the water in the vessel or the altitude of the water above the bottom of the orifice, and CF the altitude of the water above the top of the orifice. The vertical breadth of the orifice is equal to CD − CF.

154. Let x be the mean height of the fluid above the orifice, or the height due to a velocity, which, if communicated to all the particles of the issuing fluid, would make the same quantity of water issue in the time T, as if all the particles moved with the different velocities due to their different depths below the surface, then by Prop. II. the quantity discharged or \( W = 2T \times MO \times CD - CF \times \sqrt{x} \times 16.087 \), the area of the orifice being MO × CD—CF, and by making this value of W equal to its value in the preceding article, we have the following equation.

2T × MO × CD—CF × √x × 16.087 = T × √16.087 × MO × 3P/√P—4CF/√CF, which by division and reduction, and the substitution of 4P instead of CD its equal, becomes

\[ x = \frac{4(P/\sqrt{P} - 4CF/\sqrt{CF})^2}{4(\frac{4}{3}P - CF)^2} \]

Now this value of x is evidently different from the distance of the centre of gravity of the orifice from the surface of the fluid, for this distance is \( \frac{CD + CF}{2} \) or \( \frac{\frac{4}{3}P + CF}{2} \). But in proportion as CE increases, the other quantities remaining the same, the value of x will approach nearer the distance of the centre of gravity of the orifice from the surface of the fluid; for when CF becomes infinite, the parabolic arch CHE will become a straight line, and consequently the mean ordinate of the curve, which is represented by the mean velocity of the water, will pass through the middle of FD or the centre of gravity of the orifice.

PROP. IV.

155. To find the time in which a quantity of fluid equal to ABRT, will issue out of a small orifice in the side or bottom of the vessel AB, that is, the time in which the surface AB will descend to RT.

Draw DE, de at an infinitely small distance and parallel to AB. The lamina of fluid D de E may be represented by DE × ob; DE expressing the area of the surface. When the surface of the water has descended to DE, the quantity of fluid which will be discharged by an uniform velocity in the time T, will be T × √16.087 × 2A × √o m, A being the area of the orifice, as in Prop. II. But as the variation in the velocity of the water will be infinitely small, when the surface descends from DE to de, its velocity may be regarded as uniform. The time, therefore, in which the surface describes the small height ob will be found by the following analogy; T × √16.087 × 2A × √o m : T = DE × ob : \( \frac{DE \times ob}{\sqrt{16.087} \times 2A \times \sqrt{o m}} \). Now as this formula expresses the time in which the surface descends from DE to de, and as the same may be shewn of every other elementary portion of the height CS, the sum of all these elementary times will give us the value of T, the time in which the surface AB falls down to RT. For this purpose, draw GP equal and parallel to CN, and upon it as an axis, describe the parabola PVQ, having its parameter P equal to 4 GP. Continue the lines AB, DE, de, RT, so as to form the ordinates HF, hF, UV, of the parabola. Upon GP as an axis describe a second curve, so that the ordinate GM may be equal to the area of the surface at AB, divided by the corresponding ordinate GQ of the parabola, and that the ordinate H may be the quotient of the area of the surface at DE divided by the ordinate HF. Now (CONIC SECTIONS, Part I. Prop. X.) \( HF^2 = HP \times P \), or \( HF = \sqrt{HP \times P} \), that is, \( \frac{HF}{\sqrt{P}} = HP \); and since \( o m = HP \), \( \frac{DE}{\sqrt{o m}} = \frac{DE \times \sqrt{P}}{HF} \). But by the construction of the curve MN, we have \( \frac{DE}{HF} = HR \), consequently \( \frac{DE}{\sqrt{o m}} = HR \times \sqrt{P} \). The elementary time therefore, expressed by \( \frac{DE \times ob}{\sqrt{16.087} \times 2A \times \sqrt{o m}} \) will, by the different substitutions now mentioned, be \( \frac{HR \times ob \times \sqrt{P}}{2A \sqrt{16.087}} \) or \( \frac{\sqrt{P}}{2A \sqrt{16.087}} \times HR \times ob \). But the factor \( \frac{\sqrt{P}}{2A \sqrt{16.087}} \) consisting of constant quantities is itself constant, and the other factor HR × ob represents the variable curvilineal area HR × h. Now as the same may be shewn of every other element of the time T, compared with the corresponding elements of the area GU t M, it follows that the time T required, will be found by multiplying the constant quantity \( \frac{\sqrt{P}}{2A \sqrt{16.087}} \) by the curvilineal area GU t M; therefore \( T = \frac{\sqrt{P}}{\sqrt{16.087}} \times \frac{GU t M}{2A} \), and the time in which the surface descends to m n, or in which the vessel empties itself, will be equal to \( \frac{\sqrt{P}}{\sqrt{16.087}} \times \frac{GPNM}{2A} \).

COR. The quantity of fluid discharged in the given time T may be found by measuring the contents of the vessel AB between the planes AB, and RT, the descent of the surface AB, viz. the depth CS, being known.

PROP. V.

156. To find the time in which a quantity of fluid equal to ABRT will issue out of a small orifice in the side or bottom of the cylindrical vessel AB, that is, the time in which the surface AB will descend to RT.

Let us suppose that a body ascends through the height m C with a velocity increasing in the same manner as if the vessel AB were inverted, and the body fell from m to C. The velocity of the ascending body at different points of its path being proportional to the square roots of the heights described, will be expressed by the ordinates of the parabola PVQ. The line DE being infinitely near to de, as soon as the body arrives at b it will describe the small space bo or hH in a portion of time infinitely small, with a velocity represented by the ordinate HF. Now the time in which the body will ascend through the space m C or its equal PG will be \( \frac{\sqrt{PG}}{\sqrt{16.087}} \), because \( \sqrt{16.087} : \sqrt{PG} = \frac{\sqrt{PG}}{\sqrt{16.087}} \) (See MECHANICS); and if the velocity impressed Motion of impressed upon the body when at C were continued uniformly, it would run through a space equal to 2 GP or GQ in the time \( \frac{\sqrt{PG}}{\sqrt{16.087}} \). But (DYNAMICS, 22.) the times of description are as the spaces described directly, and the velocities inversely, and therefore the time of describing the space 2 GP or GQ uniformly, viz. the time \( \frac{\sqrt{PG}}{\sqrt{16.087}} \) will be to the time of describing the space hH uniformly, as, \( \frac{GQ}{GQ} : \frac{Hh}{HF} \), that is, as \( \frac{GQ}{GQ} \) or \( 1 : \frac{\sqrt{PG}}{\sqrt{16.087}} = \frac{Hh}{HF} : \frac{\sqrt{PG}}{\sqrt{16.087}} \)

\( \times \frac{Hh}{HF} \) the time in which the ascending body will describe Hh uniformly; but PG being equal to \( \frac{1}{4} P \), the parameter of the parabola, we shall have \( \sqrt{\frac{PG}{16.087}} = \frac{1}{2} \sqrt{P} \). Substituting this value of \( \sqrt{\frac{PG}{16.087}} \) in the last formula, we shall have for the expression of the time of describing Hh uniformly \( \frac{\frac{1}{2} \sqrt{P}}{\sqrt{16.087}} \times \frac{Hh}{HF} \).

But by Prop. IV. the time in which the surface DH descends into the position dh, that is, in which it describes Hh, is represented by \( \frac{\sqrt{P}}{2A\sqrt{16.087}} \times Hr \times ob \) or \( \frac{\sqrt{P}}{\sqrt{16.087}} \times \frac{Hr \times Hh}{2A} \). Therefore the time in which the ascending body moves through hH, is to the time in which the descending surface moves through Hh as \( \frac{\frac{1}{2} \sqrt{P}}{\sqrt{16.087}} : \frac{\sqrt{P}}{\sqrt{16.087}} \times \frac{Hr \times Hh}{2A} \), which expressions after being multiplied by 2, and after substituting in

\[ T = \sqrt{\frac{PG}{16.087}} \times \frac{DE}{A} \sqrt{\frac{PU}{16.087}} \times \frac{DE}{A} = \frac{DE \sqrt{PG} - DE \sqrt{PU}}{A \sqrt{16.087}} \]

\[ T = \frac{DE \times \sqrt{PG} - \sqrt{PU}}{A \sqrt{16.087}} \] Hence \[ PU = \left( \frac{T, A \sqrt{16.087}}{DE} - \sqrt{PG} \right)^2 \\ PG = \left( \frac{T, A \sqrt{16.087}}{DE} + \sqrt{PU} \right)^2 \] \[ PG - PU \text{ or } UG = \frac{2T, A \times DE \sqrt{PG} \times 16.087 - T^2 A^2 \times 16.087}{DE^2} \]

As the quantity of fluid discharged while the surface AB descends to RT is equal to DE × UG, we shall have

\[ W = DE \times \frac{2T, A \times DE \sqrt{PG} \times 16.087 - T^2 A^2 \times 16.087}{DE^2} \] \[ A = \frac{DE \times \sqrt{PG} \times \sqrt{PU}}{T \sqrt{16.087}} \] \[ DE = \frac{T, A \sqrt{16.087}}{\sqrt{PG} - \sqrt{PU}} \] HYDRODYNAMICS.

PROP. VI.

157. If two cylindrical vessels are filled with water, the time in which their surfaces will descend through similar heights will be in 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, directly, and the area of the orifices inversely.

Let ABmn, A'B'm'n' be the two vessels; then by the last proposition, the time T', in which the surface AB of the first descends to RT, will be to the time T' in which the surface A'B' of the second descends to R'T' as \( \frac{DEX \times \sqrt{PG} \times \sqrt{PU}}{A \sqrt{16.c87}} \) to \( \frac{D'E'X \times \sqrt{PG'} - \sqrt{PU'}}{A' \sqrt{16.c87}} \),

or, by dividing by \( \sqrt{16.c87} \), as \( \frac{DEX \times \sqrt{PG} - \sqrt{PU}}{A} \)

to \( \frac{D'E'X \times \sqrt{PG} - \sqrt{PU}}{A} \). Q.E.D.

158. Cor. Hence the time in which two cylindrical vessels full of water will empty themselves, will be in the compound ratio of their bases and the square roots of their altitudes directly, and the area of the orifices inversely; for in this time the surfaces AB, A'B' descend to mn, m'n' respectively, and therefore \( \sqrt{PG} - PU = \sqrt{PG} \);

<table> <tr> <th>Hours.</th> <th>0</th> <th>1</th> <th>2</th> <th>3</th> <th>4</th> <th>5</th> <th>6</th> <th>7</th> <th>8</th> <th>9</th> <th>10</th> <th>11</th> <th>12</th> </tr> <tr> <th>Difference of each Hour above the bottom.</th> <td>144</td> <td>121</td> <td>100</td> <td>81</td> <td>64</td> <td>49</td> <td>36</td> <td>25</td> <td>16</td> <td>9</td> <td>4</td> <td>1</td> <td>0</td> </tr> <tr> <th>Number of Parts in each Hour.</th> <td>23</td> <td>21</td> <td>19</td> <td>17</td> <td>15</td> <td>13</td> <td>11</td> <td>9</td> <td>7</td> <td>5</td> <td>3</td> <td>1</td> <td>0</td> </tr> </table>

For since the velocity with which the surface AB descends, the area of that surface being always the same, is as the square roots of its altitude above the orifice (Prop. I. Cor. 6.); and since the velocities are as the times of description, the times will also be as the square roots of the altitudes, that is, when

12 11 10 9 &c. are the times 144 121 100 81 will be the altitudes of the surface. Q.E.D.

PROP. VIII.

160. To explain the lateral communication of motion in fluids.

This property of fluids in motion was discovered by M. Venturi, professor of natural philosophy in the university of Modena, who has illustrated it by a variety of experiments in his work on the lateral communication of motion in fluids. Let a pipe AC, 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 while, 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 water which conveys a blast to furnaces, and which shall be blowing described in a future part of this article. The lateral machines, 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 Fig. 9. the reservoir AB 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 o n m, 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 incurved tube o n m. If the descending leg of this glass syphon be six inches and a half longer than the other, the red liquor will rise to the very top of the syphon, 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 syphon o n m, and

Motion of drags a portion along with it. The air in the syphon Fluids, &c. is therefore rarefied, and this process of rarification 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 syphon, being thus destroyed, the red liquor will rise in the syphon, 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.

PROP. IX.

161. To find the horizontal distance to which fluids will spout from an orifice perforated in the side of a vessel, and the curve which it will describe.

Let AB be a vessel filled with water, and C an orifice in its side, so inclined to the horizon as to discharge the fluid in the direction CP. If the issuing fluid were influenced by no other force except that which impels it out of the orifice, it would move with an uniform motion in the direction CP. But immediately upon its exit from the orifice C it is subject to the force of gravity, and is therefore influenced by two forces, one of which impels it in the direction CP, and the other draws it downwards in vertical lines. Make CE equal to EG, and CP double of CS the altitude of the fluid. Draw PL parallel to CK and join SL. Draw also EF, GH parallel to CN, and FM, HN parallel to CG, and let CM, CN represent the force of gravity, or the spaces through which it would cause a portion of fluid to descend in the time that this portion would move through CE, CG respectively by virtue of the impulsive force. Now, it follows from the composition of forces, (Dynamics, 135.) that the fluid at C, being solicited in the direction CE by a force which would carry it through CE in the same time that the force of gravity would make it fall through CM, will describe the diagonal CF of the parallelogram CEFM, and will arrive at F in the same time that it would have reached E by its impulsive force, or M by the force of gravity; and for the same reason the portion of the fluid will arrive at H in the same time that it would have reached G by the one force, and N by the other. The fluid therefore being continually deflected from its rectilineal direction CP by the force of gravity, will describe a curve line CEHP, which will be a parabola: for since the motion along CP must be uniform, CE, CG will be to one another as the times in which they are described; and may therefore represent the times in which the fluid would arrive at E and G, if influenced by no other force. But in the time that the fluid has described CE gravity has made it fall through EF, and in the time that it would have described CG, gravity has caused it to fall through GH. Now, since the times are as the squares of the times in which they are described, (Dynamics, 37. 2.) we shall have

\[ EF : GH = CE^2 : CG^2. \]

But on account of the parallelograms CEFM, CGHN, EF and GH are equal to CM and CN respectively, and MF, NH to CE, CG respectively; therefore \( CM : CN = MF^2 : NH^2 \), which is the property of the parabola, CM, CN being the abscissa, and MF, NH the ordinates (Conic Sections, Part I. Prop. IX. Cor.)

162. On account of the parallels LP, CX, LC, GX, the triangles LCP, GCX are similar, and therefore (Geom. Sect. IV. Theor. XX.) \( CG : CX = PC : PL \) and \( GX : CX = CL : PL \). Hence \( CG = \frac{CX \times PC}{PL} \), and \( GX = \frac{CX \times CL}{PL} \), but since \( PC = 2CS \), we have \( CG = \frac{CX \times 2CS}{PL} \), and since \( GX = GX - HX \), we shall have

\[ GH = \frac{CX \times CL}{PL} - HX. \]

But as the parameter of the parabola CRK is equal to 4 CS (1), we have, by the property of this conic section, \( NH^2 = CN \times 4CS \), or \( CG^2 = GH \times CS \); therefore, by substituting these quantities the preceding values of \( CG \) and \( GH \), we shall have

\[ CX^2 \times CS = CX \times CL \times PL - HX \times PL^2. \]

Now, it is evident, from this equation, that \( HX \) is nothing, or vanishes when \( CX = 0 \), or when \( CX = \frac{CL \times PL}{CS} \), for \( HX \) being \( = 0 \), \( HX \times PL^2 \), will also be \( = 0 \), and the equation will become \( CX^2 \times CS = CX \times CL \times PL \), or dividing by \( CX \) and \( CS \), it becomes \( CX = \frac{CL \times PL}{CS} \).

But when \( HX \) vanishes towards K, \( CX \) is equal to \( CK \), consequently \( CK = \frac{CL \times PL}{CS} \). Bifect \( CK \) in T, then \( CT = \frac{CK}{2} \), and \( CT = \frac{CL \times PL}{2CS} \). Draw TR perpendicular to \( CK \), and TR will be found \( = \frac{CL^2}{4CS} \).

Then if \( HM \) be drawn at right angles to \( HX \), we shall have \( CX = CT - HM = \frac{CL \times PL}{2CS} - HM \) and \( HX = RT - Rm = \frac{CL^2}{4CS} - Rm \). After substituting these values of \( CX \) and \( HX \) in the equation \( CX^2 \times CS = CX \times CL \times PL - HX \times PL^2 \), it will become, after the necessary reductions, \( Hm^2 = \frac{PL^2}{CS} \times Rm \). The curve CRK is

(1) The parameter of the parabola described by the issuing fluid, is equal to four times the altitude of the fluid above the orifice. For since the fluid issues at C with a velocity equal to that acquired by falling through SC, if this velocity were continued uniform, the fluid would move through 2 CS or CP, in the same time that a heavy body would fall through SC. Draw PQ parallel to CS, and QW to CP; then since Q is in the parabola, the fluid will describe CP uniformly in the same time that it falls through CW by the force of gravity, therefore \( CW = CS \). Now \( CP = 2CS \), and \( CP^2 = 4CS^2 = 4 \times CS \times CS = 4 \times CS \times CW \); but it is a property of the parabola, that the square of the ordinate WQ or CP is equal to the product of the abscissa CW and the parameter, therefore 4 CS is the parameter of the parabola. Motion of is therefore a parabola whose vertex is R, its axis RT and its parameter \( \frac{PL^3}{CS} \), R m being an abscissa of the axis, and H m its correspondent ordinate. Now, making \( a = CS \), the altitude of the reservoir; R = radius; \( m = PL \) the sine of the angle PCL; and \( n = CL \), the cosine of the same angle, CP being radius. Then CP : PL = R : m, therefore PL × R = CP × m, and dividing by R and substituting 2 a or 2 CS instead of its equal CP, we have \( PL = \frac{2a\ m}{R} \), and by the very same reasoning, we have \( CL = \frac{2\ a\ n}{R} \). Hence \( RT = \frac{CL^2}{4\ CS} \) will be \( = \frac{4a^2n^2}{R^2} \) divided by 4 a, or \( RT = a \times \frac{n^2}{R^2} \), and \( CT = \frac{CL \times PL}{2\ CS} = \frac{4a^2m\ n}{2a \times R^2} = 2\ a \times \frac{mn}{R^2} \), and the parameter of the parabola \( = \frac{PL^3}{CS} = \frac{4a^2m^2}{a \times R^2} = 4a \times \frac{m^2}{R^2} \).

163. Hence we have the following construction. With \( \frac{3}{2} CS \) as radius, describe the semicircle 8GC, which the direction CR of the jet or issuing fluid meets in G. Draw GN perpendicular to CS, and having prolonged it towards R, make GR equal to GN. From R let fall RT perpendicular to CK and meeting it in T, and upon RT, CT describe the parabola CRK having its vertex in R, this parabola shall be the course of the issuing fluid. For by the construction NR or CT = 2 GN, and on account of the similar triangles SGC, CGN, SC : SG = CG : GN; hence SC × GN = SG × CG, or \( 2\ GN, \) or \( CT = \frac{2\ SG \times CG}{SC} \). But from the similarity of triangles CS : CG = SG : GN and CS : CG = CG : CN, consequently, when CG is radius or = R, GN will be the sine m of the angle GCS, and CN its cosine n; and we shall then have, by Euclid VI. 16, and reduction \( SG = \frac{CS \times m}{R} \), and \( CG = \frac{CS \times n}{R} \). By substituting these values of SG and CG in the equation \( CT = \frac{2\ SG \times CG}{SC} \), we have \( CT = \frac{2}{SC} \times \frac{CS \times m}{R} \times \frac{CS \times n}{R} = \frac{2\ CS \times m \times CS \times n}{CS \times R \times R} = \frac{2\ CS \times mn}{R^2} = 2\ a \times \frac{mn}{R^2} \). But the parameter P of the parabola CRK is equal to \( \frac{CT^3}{RT^2} \), because it is a third proportional to the abscissa and its ordinate, therefore \( P = \frac{4a^2 \times m^3 n^2}{R^2 \times RT^2} \). Now RT = CN, and \( CN = \frac{NG \times n}{m} \), because CN : NG = m : n, or CN \( = RT = a \times \frac{n^2}{R^2} \) by substituting the preceding value of NG.

Therefore the parameter \( P = \left( \frac{4a^2 \times m^3 n^2}{R^4} \right) \div \left( \frac{a \times n^2}{R^2} \right) = 4 \times \frac{m^2}{R^2} \), which is the same value of the parameter as was found in the preceding article, and therefore verifies the construction.

164. Cor. 1. Since NG = GR and CT = TK, the amplitude or distance CK, to which the fluid will reach on a horizontal plane, will be 4 NG, or quadruple the sine of the angle formed by the direction of the jet and a vertical line, the chord of the arch CG, being radius.

165. Cor. 2. If Sn be made equal to CN, and ng be drawn parallel to CT, and gr be made equal to ng; then if the direction of the jet be Cg, the fluid will describe the parabola Cr K whose vertex is r, and will meet the horizontal line in K, because ng = NG, and \( 4ng = 4NG = CK \). The same may be shewn of every other pair of parabolas whose vertices Rr are equidistant from a c a horizontal line passing through the centre of the circle.

166. Cor. 3. Draw the ordinate ab through the centre a, and since this is the greatest ordinate that can be drawn, the distance to which the water will spout, being equal to 4 a, will be the greatest when its line of direction passes through b, that is, when it makes an angle of 45° with the horizon.

167. Cor. 4. If an orifice be made in the vessel AB at N, and the water issues horizontally in the direction NG, it will describe the parabola NT, and CT will be equal to 2 NG. For (by Prop. IX. note) the parameter of the parabola NT is equal to 4 NS, and by the property of the parabola \( \overline{CT} = NC \times 4NS \), or \( \frac{1}{2} CT = 2\sqrt{NC \times NS} \); but by the property of the circle (GEOM. Sect. IV. Theor. XXVIII.) \( NG^2 = NC \times NS \), and \( NG = \sqrt{NC \times NS} \), hence \( CT = 2NG \). If the fluid is discharged from the orifice at n, so that Sn = CN, ng will be = NG, and it will spout to the same distance CT.

Prop. X.

168. 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 EFfe. 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 m n 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 former. Therefore d being the diameter of the vena contracta, and d that of the pipe CD, the area of the one will be to the area of the other, as \( d^2 : d^3 \), (GEOMETRY, Sect. VI. Prop. IV.) consequently we shall have \( d^2 : d^3 = \sqrt{A} : \frac{d^3 \sqrt{A}}{d^2} \), the velocity of the water in the pipe. But since the velocity \( \sqrt{A} \) is due to the altitude A, the velocity \( \frac{d^3 \sqrt{A}}{d^2} \) will be due to the altitude \( \frac{d^3 A}{d^2} \). 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 \( \frac{d^3 \sqrt{A}}{d^2} \), the extremity D n of the pipe will sustain a pressure equal to the difference of the pressures produced by the velocities \( \sqrt{A} \) and \( \frac{d^3 \sqrt{A}}{d^2} \), that is, by a pressure \( A - \frac{3^4 A}{d^4} \), \( A \) representing the pressure which produces the velocity \( \sqrt{A} \), and \( \frac{3^4 A}{d^4} \) the pressure which produces the velocity \( \frac{\sqrt{A}}{d^2} \). But this pressure is distributed through every part of the pipe CD, consequently the pressure sustained by the sides of the pipe will be \( A - \frac{3^4 A}{d^4} \).

169. 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 \( A - \frac{3^4 A}{d^4} \). When the diameter \( d \) of the orifice is equal to the diameter \( d \) of the pipe, the altitude becomes \( A - A \) 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.

170. 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 orifice may be easily found. Let W be the quantity of water discharged in a given time by the proposed aperture under the pressure \( A \), and let w be the quantity discharged under the pressure \( A - \frac{3^4 A}{d^4} \). Then W:

\[ w = \sqrt{A} : \sqrt{A - \frac{3^4 A}{d^4}}, \text{ consequently, } w \times \sqrt{A} = W \times \sqrt{A - \frac{3^4 A}{d^4}} \]

\[ w = \frac{W \times \sqrt{A - \frac{3^4 A}{d^4}}}{\sqrt{A}} = W \sqrt{\frac{d^4 - 3^4}{d^4}} \]

Therefore, since W may be determined by the experiments in the following chapter, w is known.

CHAP. II. Account of Experiments on the Motion of Water discharged from vessels, either by Orifices or additional Tubes, or running in Pipes or open Canals.

171. In the preceding chapter, we have taken notice of the contraction produced upon the vein of fluid issuing from an orifice in a tin plate, and have endeavoured to ascertain its cause. According to Sir Isaac Newton, the diameter of the vena contracta is to that of the orifice as 21 to 25. Polemus makes it as 11 to 13; Bernouilli as 5 to 7; the Chevalier de Buat as 6 to 9; Boffut as 41 to 50; Michelotti, as 4 to 5; and Venturi, as 4 to 5. This ratio, however, is by no means constant. It varies with the form and position of the orifice, with the thickness of the plate in which the orifice is made, and likewise with the form of the vessel and the weight of the superincumbent fluid. But these variations are too trifling to be regarded in practice.—We shall now lay before the reader an account of the results of the experiments of different philosophers, but particularly those of the Abbé Boffut, to whom the science is deeply indebted both for the accuracy and extent of his labours.

SECT. I. On the Quantity of Water discharged from Vessels constantly full by Orifices in thin Plates.

172. In the following experiments, which were frequently repeated in various ways, the orifice was pierced in a plate of copper about half a line thick. When the orifice is in the bottom of the vessel, it is called a horizontal orifice, and when it is in the side of it, it is called a lateral orifice.

<table> <tr> <th>Altitude of the fluid above the centre of the orifice.</th> <th>Form and position of the orifice.</th> <th>The orifice's diameter.</th> <th>N° of cub. in. discharged in a minute.</th> </tr> <tr> <td rowspan="2">Ft. In. Lin</td> <td>Circular and Horizontal</td> <td>6 lines</td> <td>2311</td> </tr> <tr> <td>Circular and Horizontal</td> <td>1 inch</td> <td>9281</td> </tr> <tr> <td></td> <td>Circular and Horizontal</td> <td>2 inches</td> <td>37203</td> </tr> <tr> <td></td> <td>Rectangular and Horizontal</td> <td>1 inch by 3 lines</td> <td>2933</td> </tr> <tr> <td rowspan="2">9 0 0</td> <td>Horizontal and Square</td> <td>1 inch, side</td> <td>11817</td> </tr> <tr> <td>Horizontal and Square</td> <td>2 inch, side</td> <td>47361</td> </tr> <tr> <td></td> <td>Lateral and Circular</td> <td>6 lines</td> <td>2018</td> </tr> <tr> <td></td> <td>Lateral and Circular</td> <td>1 inch</td> <td>8135</td> </tr> <tr> <td rowspan="2">4 0 0</td> <td>Lateral and Circular</td> <td>6 lines</td> <td>1353</td> </tr> <tr> <td>Lateral and Circular</td> <td>1 inch</td> <td>5436</td> </tr> <tr> <td>5 0 7</td> <td>Lateral and Circular</td> <td>1 inch</td> <td>628</td> </tr> </table>

173. 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 or 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

\[ W : w = A : a, \text{ and by the second} \]

\[ w : W = \sqrt{H} : \sqrt{h}. \text{ Multiplying these analogies together, gives us} \]

\[ W w : W' w = A \sqrt{H} : a \sqrt{h}, \text{ and dividing by } w, \]

\[ W : W' = A \sqrt{H} : a \sqrt{h}. \] 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 circumference 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 augments, 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.

174. The abbé Bollat has given the following table containing a comparison of the theoretical with the real discharges, for an orifice one inch 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. The fourth column was computed by M. Prony.

<table> <tr> <th>Constant altitude of the water in the reservoir above the centre of the orifice.</th> <th>Theoretical discharges through a circular orifice one inch in diameter.</th> <th>Real discharges in the same time through the same orifice.</th> <th>Ratio of the theoretical to the real discharges.</th> </tr> <tr> <th>Paris Feet</th> <th>Cubic Inches</th> <th>Cubic Inches</th> <th></th> </tr> <tr> <td>1</td> <td>4381</td> <td>2722</td> <td>1 to 0.62133</td> </tr> <tr> <td>2</td> <td>6196</td> <td>3846</td> <td>1 to 0.62733</td> </tr> <tr> <td>3</td> <td>7389</td> <td>4710</td> <td>1 to 0.6264</td> </tr> <tr> <td>4</td> <td>8763</td> <td>5436</td> <td>1 to 0.62034</td> </tr> <tr> <td>5</td> <td>9797</td> <td>6075</td> <td>1 to 0.62010</td> </tr> <tr> <td>6</td> <td>10732</td> <td>6654</td> <td>1 to 0.62000</td> </tr> <tr> <td>7</td> <td>11592</td> <td>7183</td> <td>1 to 0.61965</td> </tr> <tr> <td>8</td> <td>12292</td> <td>7672</td> <td>1 to 0.61911</td> </tr> <tr> <td>9</td> <td>13144</td> <td>8135</td> <td>1 to 0.61892</td> </tr> <tr> <td>10</td> <td>13855</td> <td>8574</td> <td>1 to 0.61883</td> </tr> <tr> <td>11</td> <td>14330</td> <td>8992</td> <td>1 to 0.61873</td> </tr> <tr> <td>12</td> <td>15180</td> <td>9384</td> <td>1 to 0.61810</td> </tr> <tr> <td>13</td> <td>15797</td> <td>9764</td> <td>1 to 0.61810</td> </tr> <tr> <td>14</td> <td>16393</td> <td>10130</td> <td>1 to 0.61793</td> </tr> <tr> <td>15</td> <td>16968</td> <td>10472</td> <td>1 to 0.61716</td> </tr> </table>

175. 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. Thus, if we take the altitudes 1 and 4, whose square roots are as 1 to 2, the real discharges taken from the table are 2722, 5436, which are to one another very nearly as 1 to 2, their real ratio being as 1 to 1.997.

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.

176. 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. 173. 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 16918 cubic inches in a minute, we shall have this analogy, 1/4 : 15 : 01 : 30 = 16968 : 215061 cubic inches, the quantity required. This quantity being diminished in the ratio of 1 to .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. 173.) 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.

177. As the velocity of falling bodies is 16.087 feet per second, the velocity due to 16.087 feet will be 32.174 feet per second, and as the velocities are as the square roots of the height, we shall have \( \sqrt{16.087} : \sqrt{H} = 32.174 : V \) the velocity due to any other height, consequently \( V = \frac{32.174 \sqrt{H}}{\sqrt{16.087}} = \frac{32.174 \sqrt{H}}{4.011} = 8.016 \sqrt{H} \), 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.

178. According to the experiments of M. Eytlwein, published at Berlin in 1801, in his treatise Handbuch der Mechanik und der Hydraulik, the following experiments are the ratios between the theoretical and actual discharges, and the coefficients by which the height may be multiplied in order to find the velocities of the issuing fluid.

TABLE III. Results of Eytelwein's Experiments.

<table> <tr> <th>No</th> <th>Nature of the orifices employed.</th> <th>Ratio between the theoretical and real discharges.</th> <th>Coefficients for finding the velocities.</th> </tr> <tr> <td>1</td> <td>When the orifice has the form of the contracted stream</td> <td>1 to 0.973</td> <td>7.8</td> </tr> <tr> <td>2</td> <td>For wide openings whose bottom is on a level with that of the reservoir</td> <td>1 to 0.961</td> <td>7.7</td> </tr> <tr> <td>3</td> <td>For flues with walls in a line with the orifice</td> <td>1 to 0.961</td> <td>7.7</td> </tr> <tr> <td>4</td> <td>For bridges with pointed piers</td> <td>1 to 0.961</td> <td>7.7</td> </tr> <tr> <td>5</td> <td>For narrow openings whose bottom is on a level with that of the reservoir</td> <td>1 to 0.861</td> <td>6.9</td> </tr> <tr> <td>6</td> <td>For smaller openings in a flue with side walls</td> <td>1 to 0.861</td> <td>6.9</td> </tr> <tr> <td>7</td> <td>For abrupt projections and square piers of bridges</td> <td>1 to 0.861</td> <td>6.9</td> </tr> <tr> <td>8</td> <td>For openings in flues without side walls</td> <td>1 to 0.635</td> <td>5.1</td> </tr> <tr> <td>9</td> <td>For orifices in a thin plate</td> <td>1 to 0.625</td> <td>5.0</td> </tr> </table>

179. M. Eytelwein has likewise shown, that the quantity of water discharged from rectangular orifices in the side of a reservoir extending to the surface, may be found by taking two-thirds of the velocity due to the mean height, and allowing for the contraction according to the form of the orifice.

SECT. II. On the Quantity of Water discharged from Vessels constantly full, by small Tubes adopted to Circular Orifices.

180. The difference between the natural 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 unguents: 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.

181. If a cylindrical tube two inches long, and two inches in diameter, be inserted in the reservoir, and if this orifice is 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 one inch in diameter, and two inches long, the water followed the sides of the tube, and the vein of fluid ceased to contract. While M. Bofluit was repeating this experiment, he prevented the escape of the fluid by placing the instrument MN, 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 one inch fix 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 one inch; and when it was so small as half an inch, the water uniformly detached itself from its circumference, and formed the vena contracta.

182. TABLE IV. Shewing the Quantities of Water discharged by Cylindrical Tubes one inch in diameter with different lengths.

<table> <tr> <th></th> <th>Variable lengths of the tubes expressed in lines.</th> <th>cubic inches discharged in a minute.</th> <th>Quantities of fluid discharged from cylindrical tubes of the same diameters but different lengths.</th> </tr> <tr> <td>Constant altitude of the fluid above the superior base of the tube being 11 feet 8 inches and 10 lines.</td> <td>The tube being filled with the issuing fluid</td> <td>48 12274<br>24 12183<br>18 12168</td> <td></td> </tr> <tr> <td></td> <td>The tube not filled with the issuing fluid</td> <td>18 9282</td> <td></td> </tr> </table>

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 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 vena 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 con- clude 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.

183. In order to determine the effect of tubes of different diameters, under different altitudes of water in the reservoir, M. Bofut instituted the experiments, the results of which are exhibited in the following table.

<table> <tr> <th rowspan="2">Quantities of water discharged by cylindrical tubes of the same length but different diameters.</th> <th colspan="2">Diameter of the tube.</th> <th rowspan="2">Quantity of water discharged in a minute.</th> </tr> <tr> <th>Const. altitude of the water above the orifice.</th> <th>Diam. inches</th> </tr> <tr> <td rowspan="4">3 10</td> <td>The tube being filled with the issuing fluid.</td> <td>6</td> <td>1689</td> </tr> <tr> <td>The tube not filled with the issuing fluid.</td> <td>10</td> <td>4703</td> </tr> <tr> <td>The tube being filled with the issuing fluid.</td> <td>6</td> <td>1293</td> </tr> <tr> <td>The tube not filled with the issuing fluid.</td> <td>10</td> <td>3598</td> </tr> <tr> <td rowspan="3">2 0</td> <td>The tube being filled with the issuing fluid.</td> <td>6</td> <td>1222</td> </tr> <tr> <td>The tube not filled with the issuing fluid.</td> <td>10</td> <td>3402</td> </tr> <tr> <td>The tube not filled with the issuing fluid.</td> <td>6</td> <td>935</td> </tr> <tr> <td></td> <td></td> <td>10</td> <td>2603</td> </tr> </table>

184. 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 diameter. 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.

185. 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.

<table> <tr> <th>Const. altitude of the water in the reservoir above the centre of the orifice.</th> <th>Theoretical discharges through a circular orifice one inch in diameter.</th> <th>Real discharges in the same time by a cylindrical tube one inch in diameter and two inches long.</th> <th>Ratio of the theoretical to the real discharges.</th> </tr> <tr> <th>Paris Feet.</th> <th>Cubic inches.</th> <th>Cubic inches.</th> <th></th> </tr> <tr> <td>1</td> <td>4381</td> <td>3539</td> <td>1 to 0.81781</td> </tr> <tr> <td>2</td> <td>6196</td> <td>5002</td> <td>1 to 0.82729</td> </tr> <tr> <td>3</td> <td>7589</td> <td>6126</td> <td>1 to 0.80724</td> </tr> <tr> <td>4</td> <td>8763</td> <td>7270</td> <td>1 to 0.86681</td> </tr> <tr> <td>5</td> <td>9797</td> <td>7920</td> <td>1 to 0.86638</td> </tr> <tr> <td>6</td> <td>10732</td> <td>8654</td> <td>1 to 0.86638</td> </tr> <tr> <td>7</td> <td>11592</td> <td>9340</td> <td>1 to 0.85773</td> </tr> <tr> <td>8</td> <td>12392</td> <td>9975</td> <td>1 to 0.84966</td> </tr> <tr> <td>9</td> <td>13144</td> <td>10579</td> <td>1 to 0.80485</td> </tr> <tr> <td>10</td> <td>13855</td> <td>11151</td> <td>1 to 0.80483</td> </tr> <tr> <td>11</td> <td>14330</td> <td>11693</td> <td>1 to 0.82477</td> </tr> <tr> <td>12</td> <td>15180</td> <td>12205</td> <td>1 to 0.80493</td> </tr> <tr> <td>13</td> <td>15797</td> <td>12699</td> <td>1 to 0.83390</td> </tr> <tr> <td>14</td> <td>16393</td> <td>13177</td> <td>1 to 0.80382</td> </tr> <tr> <td>15</td> <td>16968</td> <td>13620</td> <td>1 to 0.80270</td> </tr> <tr> <td>1</td> <td>2</td> <td>3</td> <td>4</td> </tr> </table>

By comparing the preceding table with that in art. 174. 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.

186. 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 2 5 feet. In order to resolve this question, find (by art. 176.) 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.

187. Hitherto we have supposed the tube to be exactly cylindrical. When its interior surface, however, is conical, the quantities discharged undergo a considerable variation, which may be estimated from the following experiments of the marquis Poleni, published in his work De Cessibus per quae derivantur fluviorum aquas, &c. which appeared at Padua in 1718. TABLE VII. Shewing the Quantities of Water discharged by Conical Tubes of different Diameters.

<table> <tr> <th rowspan="2">Apertures Employed.</th> <th colspan="2">Interior diameter.</th> <th colspan="2">Exterior diameter.</th> <th rowspan="2">Quantity discharged in a min. in cubic ft.</th> <th rowspan="2">Time in which inches were discharged.</th> </tr> <tr> <th>Conflant altitude of the water in the reservoir, 92 lines, or 1 foot 9 inches and 4 lines.</th> <th>Length of each tube</th> <th>Orifice in a thin plate.</th> <th>Cylindrical tube,</th> <th>1st Conical tube,</th> <th>2d Conical tube,</th> <th>3d Conical tube,</th> <th>4th Conical tube,</th> </tr> <tr> <td></td> <td>26 lines</td> <td>26 lines</td> <td>15877</td> <td>4' 36"</td> </tr> <tr> <td></td> <td>26</td> <td>26</td> <td>23434</td> <td>3' 7"</td> </tr> <tr> <td></td> <td>33</td> <td>26</td> <td>24758</td> <td>2' 57"</td> </tr> <tr> <td></td> <td>42</td> <td>26</td> <td>24619</td> <td>2' 58"</td> </tr> <tr> <td></td> <td>60</td> <td>26</td> <td>24345</td> <td>3' 6"</td> </tr> <tr> <td></td> <td>118</td> <td>26</td> <td>23087</td> <td>3' 5"</td> </tr> </table>

From these experiments we are authorized to conclude, 1. That the real discharges are less than those deduced from theory, which in the present case is 27425 cubic inches in a minute, and 2. That when the interior orifice of the tube is enlarged to a certain degree, the quantity discharged is increased; but that when this enlargement is too great, a contraction takes place without the exterior orifice, and the quantity discharged suffers a diminution. If the smallest base of the conical tube be inserted in the side of the reservoir, it will furnish more water than a cylindrical tube whose diameter is equal to the smallest diameter of the conical tube; for the divergency of its sides changes the oblique motion which the particles would otherwise have had, when passing from the reservoir into the tube.

188. The experiments of Poleni and Bossut having been made only with tubes of a conical and cylindrical form, M. Venturi was induced to institute a set of experiments, in which he employed tubes of the various forms exhibited in fig. 4. The results of his researches of various are contained in the following table, for which we have computed the column containing the number of cubic inches discharged in one minute, in order that the experiments of the Italian philosopher may be more easily compared with those which are exhibited in the preceding tables. The conflant 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, unless the contrary be expressed.

TABLE VIII. Shewing the Quantities of Water discharged from Orifices of various forms, the conflant Altitude of the Fluid being 32.5 French, or 34.642 English inches.

<table> <tr> <th>N°</th> <th>Nature and dimensions of the tubes and orifices.</th> <th>Time in which 4 Paris cub. ft. were discharged</th> <th>Paris cubic inches discharged in a minute.</th> </tr> <tr> <td>1</td> <td>A simple circular orifice in a thin plate, the diameter of the aperture being 1.6 inches,</td> <td>41</td> <td>10115</td> </tr> <tr> <td>2</td> <td>A cylindrical tube 1.6 inches in diameter, and 4.8 inches long,</td> <td>31</td> <td>13378</td> </tr> <tr> <td>3</td> <td>A tube similar to B, figure 4, which differs from the preceding only in having the contraction in the shape of the natural contracted vein,</td> <td>31</td> <td>13378</td> </tr> <tr> <td>4</td> <td>The short conical adutage, A, figure 4, being the first conical part of the preceding tube,</td> <td>42</td> <td>9874</td> </tr> <tr> <td>5</td> <td>The tube D, figure 4, being a cylindrical tube adapted to the small conical end A, m n being 3.2 inches long,</td> <td>42.5</td> <td>9758</td> </tr> <tr> <td>6</td> <td>The same adutage, m n being 12.8 inches,</td> <td>45</td> <td>9216</td> </tr> <tr> <td>7</td> <td>The same adutage, m n being 25.6 inches,</td> <td>48</td> <td>8640</td> </tr> <tr> <td>8</td> <td>The tube C, consisting of the cylindrical tube of Exp. 2, placed over the conical part of A,</td> <td>32.5</td> <td>12760</td> </tr> <tr> <td>9</td> <td>The double conical pipe E, ab=ac=1.6 inches, cd=0.977 inches, ef=1.376 inches, and the length c e of the outer cone =4.351 inches,</td> <td>27.5</td> <td>15081</td> </tr> <tr> <td>10</td> <td>The tube F, consisting of a cylindrical tube 3.2 inches long, and 1.376 inches in diameter, interposed between the two conical parts of the preceding,</td> <td>28.5</td> <td>14516</td> </tr> </table>

189. These experiments of Venturi inform us of a curious fact, extremely useful to the practical hydraulist. They incontestably prove, that when water is conveyed through a straight cylindrical pipe of an unlimited length, the discharge of water may be increased only by altering the form of the terminations of the pipe, that is, by making the end of the pipe A of the same form as the vena contracta, and by forming the other extremity BC into a truncated cone, having its length BC about 9 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 24.

190. M. Venturi also found, that the quantities of water discharged out of a straight tube, a curved tube forming a quadrantal 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 finuofitts 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.

191. It appears from the researches of Eytelwein, that when the shortest tube that will make the water follow its sides is applied to the reservoir, the quantity discharged will be to that deduced from theory, as 0.810 to 1.000, and the multiplier for finding the velocity will be 6.5. When the lengths of the tubes are increased from two to four times their diameter, the ratio of the actual and theoretical discharge will be 0.822 to 1.020, and the constant multiplier for finding the velocity will be 6.6. In employing a conical tube approaching to the figure of the vena contracta, the ratio of the discharges was as 0.92 to 1.00, and when its edges were rounded off, as 0.98 to 1.00 computing from its least section. He found also that the smallest quantity of water was discharged, when the interior extremity of the tube projected within the reservoir, the quantity furnished in this case being reduced to one half of what was discharged when the tube had its proper position.

192. 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 expense 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.

193. Of all the tubes that can be employed for discharging water, that is the most advantageous which has the form of a contracted vein. Hence, it will be a truncated cone with its greatest base next the reservoir, having its length equal to half the diameter of that base, and the area of the two orifices as 8 to 5, or their diameters in the subduplicate ratio of these numbers, viz. as \( \sqrt{8} : \sqrt{5} \).

SECT. III. Experiments on the Exhaustion of Vessels.

194. It is almost impossible to determine the exact time in which any vessel of water is completely exhausted. When the surface of the fluid has descended within a few inches of the orifice, a kind of conoidal funnel 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 Boffut has determined the times in which the superior surface of the fluid descends through a certain vertical height, and his results will be found in the following table.

<table> <tr> <th>Primitive altitude of the water in the vessel.</th> <th>Constant area of a horizontal section of the vessel.</th> <th>Diameter of the circular orifice.</th> <th>Depression of the upper surface of the fluid.</th> <th>Time in which this depression takes place.</th> </tr> <tr> <td>Paris Feet.</td> <td>Square Feet.</td> <td>Inch.</td> <td>Feet.</td> <td>Min. Sec.</td> </tr> <tr> <td rowspan="3">11.6666</td> <td rowspan="3">9</td> <td>1</td> <td>4</td> <td>7 25 1/2</td> </tr> <tr> <td>2</td> <td>4</td> <td>1 52</td> </tr> <tr> <td>1</td> <td>9</td> <td>20 24 1/2</td> </tr> <tr> <td></td> <td></td> <td>2</td> <td>9</td> <td>5 6</td> </tr> </table>

PG DE A PG—PU T

\( \frac{A}{\sqrt{16.087}} \), in which DE is the area of a section of the vessel, PG the primitive altitude of the surface above the centre of the orifice, PU the altitude of the surface after the time T is elapsed, A the area of the orifice, and 16.087 the space through which a heavy body descends in one second of time. That the preceding formula may be corrected, we must substitute \( 0.62 \) or \( \frac{5A}{8} \), instead of A, in the formula, 0.62, A being the area of the vena contracta; and as the measures in the preceding table are in Paris feet, we must use 15.085 instead of 16.087, the former being the distance in Paris feet, and the latter the distance in English feet, which falling bodies describe in a second. The formula, therefore, will become \( T = \frac{DE \times \sqrt{PG - PU}}{0.62A \sqrt{15.085}} \), and when the computations are made for the different diameters of the orifices and the different depressions of the fluid surface, the results will be had, which are exhibited in the last column of the following table, containing the values of T, according to theory and experience.

<table> <tr> <th>Diameter of the circular orifice.</th> <th>Depression of the upper surface of the fluid.</th> <th>Time of the depression of the surface by experiment.</th> <th>Time of the depression of the surface by the formula.</th> <th>Difference between the theory and the experiments.</th> </tr> <tr> <td>Inches.</td> <td>Feet.</td> <td>Min. Sec.</td> <td>Min. Sec.</td> <td>Seconds.</td> </tr> <tr> <td>1</td> <td>4</td> <td>7 25 1/2</td> <td>7 22 36</td> <td>3.14</td> </tr> <tr> <td>2</td> <td>4</td> <td>1 52</td> <td>1 50.59</td> <td>1.41</td> </tr> <tr> <td>1</td> <td>9</td> <td>20 24 1/2</td> <td>20 16</td> <td>8.50</td> </tr> <tr> <td>2</td> <td>9</td> <td>5 6</td> <td>5 4</td> <td>2.00</td> </tr> </table> It appears from this table 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. This may arise from 0.62 being too great a multiplier for finding the corrected diameter of the orifice.—When the orifices are in the sides of the reservoir, the altitude PG, PU of the surface may be reckoned from the centre of gravity of the orifice, unless when it is very large.

Sect. IV. Experiments on Vertical and Oblique Jets.

196. 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 Bouffut, 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.

<table> <tr> <th rowspan="2">Plate CCLXVIII fig. 6.</th> <th rowspan="2">Diameter of the horizontal tubes m P, n R, each being fix feet long.</th> <th rowspan="2">Form of the orifices.</th> <th rowspan="2">References to Fig. 6.</th> <th rowspan="2">Diameter of the orifice.</th> <th colspan="2">Altitude of the jet when rising vertically, reckoning from m.</th> <th colspan="2">Altitude of the jet when inclined a little to the vertical.</th> <th rowspan="2">Description of the jets.</th> </tr> <tr> <th>Feet. Inch</th> <th>Lines.</th> <th>Feet. Inch</th> <th>Lines.</th> </tr> <tr> <td rowspan="3">Inch. Lines.</td> <td>3 8</td> <td>Simple orifice</td> <td>H</td> <td>2</td> <td>10 0</td> <td>10</td> <td>10 4</td> <td>6</td> <td>The vertical jet beautiful.</td> </tr> <tr> <td>3 8</td> <td>———</td> <td>G</td> <td>4</td> <td>10 5</td> <td>10</td> <td>10 7</td> <td>6</td> <td>The vertical jet beautiful, not much enlarged at the top.<br>All the jets occasionally rise to different heights. This very perceptible in the present experiment.<br>The vertical jet much enlarged at top. The inclined one less so, and more beautiful.</td> </tr> <tr> <td>3 8</td> <td>———</td> <td>F</td> <td>8</td> <td>10 6</td> <td>6</td> <td>10 8</td> <td>0</td> <td>The vertical jet beautiful.</td> </tr> <tr> <td rowspan="4">Inch. Lines.</td> <td>3 8</td> <td>Conical tube</td> <td>E</td> <td>94 by 70</td> <td>9</td> <td>6</td> <td>4</td> <td>9 8</td> <td>6</td> <td>The vertical jet beautiful.</td> </tr> <tr> <td>3 8</td> <td>Cylindrical tube</td> <td>D</td> <td>4 by 70</td> <td>9</td> <td>1</td> <td>6</td> <td>7 3</td> <td>6</td> <td>The vertical jet beautiful.</td> </tr> <tr> <td>0 9½</td> <td>Simple orifice</td> <td>M</td> <td>2</td> <td>9</td> <td>11</td> <td>0</td> <td>———</td> <td>———</td> <td>The jet beautiful.</td> </tr> <tr> <td>0 9½</td> <td>———</td> <td>L</td> <td>4</td> <td>9</td> <td>7</td> <td>10</td> <td>———</td> <td>———</td> <td>The jet much deformed, and very much enlarged at top.<br>The column much broken; and the successive jets are detached from each other.</td> </tr> <tr> <td>0 9½</td> <td>———</td> <td>K</td> <td>8</td> <td>7</td> <td>10</td> <td>0</td> <td>———</td> <td>———</td> <td></td> </tr> </table>

197. It appears, from the three first experiments of the preceding table, that great jets rise higher than small ones; and from the three last 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 be the diameter of the tube, d that of the adjutage, a the altitude B m 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 (art. 150, note) 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 the same letters in the Greek character, viz. \( \Delta_1, \delta, \alpha, \beta \), and we shall have the equation \( \beta = \frac{\delta^2 \sqrt{\alpha}}{\Delta^2} \). If we wish, therefore, that the two jets be furnished in the same manner, 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 \( \frac{d^2 \sqrt{a}}{D^2} = \frac{\delta^2 \sqrt{\alpha}}{\Delta^2} \). Hence \( D^2 : \Delta^2 = d d \sqrt{a} : \delta \delta \sqrt{a} \), 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.

Experiments on the Motion of Fluids.

Now, it appears from the experiments of M. riotte (Traité de mouvement des eaux), that when the altitude of the reservoir is 16 feet, and the diameter of the adjutage fix lines, the diameter of the horizontal tube ought to be 28 lines and a half. 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 three last experiments, that the jets rise to the smaller height when the adjutage is a cylindrical tube (see D fig. 6.), 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.

198. By comparing the preceding experiments with those of Mariotte, it appears, that the differences between the heights of vertical jets, and the heights of the reservoir, are nearly as the squares of the heights of the jets. Thus, \( ab : cd = Eb^2 : Fd^2 \); therefore, if \( ab \) be known by experiment, we shall have \( cd = \frac{ab \times Fd^2}{Eb^2} \), and by adding \( cd \) to \( Fd \), we shall have the altitude of the reservoir. But if \( Fc \) were given, and it were required to find \( Fd \), the height of the jet, we have, by the preceding analogy, \( Fd^2 = \frac{Eb^2 \times cd}{ab} \). But \( cd \) is an unknown quantity, and is equal to \( Fc - Fd \), therefore, by substitution, \( Fd^2 = \frac{Eb^2 \times Fc - Fd}{ab} \), or \( Fd^2 \times \frac{Eb^2}{ab} \times \frac{Fc}{Fd} = \frac{Eb^2 \times Fc}{ab} \), which is evidently a quadratic equation, which, after reduction, becomes \( Fd = \sqrt{\frac{Eb^2 \times Fc + Eb^4}{ab} - \frac{Eb^4}{4}} \).

199. From a comparison of the 5th and 6th 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 vertical (k); 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 subfides, 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 by some philosophers 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; then 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 (art. 150. note). 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 Torricellius*, who seems to ascribe the diminution in the altitude of the jet to the gravity of the descending particles.

200. The following table exhibits all that is necessary in the formation of jets. The two first columns are taken from Mariotte†, and shew 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 fix lines in diameter. The fourth column, computed from the hypothesis in art. p. 393, contains the diameters of the horizontal tubes for an adjutage fix lines in diameter, relative to the altitudes in the second column. The thickness of the horizontal tubes will be determined in a subsequent section.

<table> <tr> <th>Altitude of the jet.</th> <th>Altitude of the reservoir.</th> <th>Quantity of water discharged in a minute from an adjutage 6 lines in diam.</th> <th>Diameters of the horizontal tubes suited to the two preceding columns.</th> </tr> <tr> <th>Paris Feet.</th> <th>Fect. Inches.</th> <th>Paris Pints.</th> <th>Lines.</th> </tr> <tr><td>5</td><td>5 1</td><td>32</td><td>21</td></tr> <tr><td>10</td><td>10 4</td><td>45</td><td>26</td></tr> <tr><td>15</td><td>15 9</td><td>56</td><td>28</td></tr> <tr><td>20</td><td>21 4</td><td>65</td><td>31</td></tr> <tr><td>25</td><td>27 1</td><td>73</td><td>33</td></tr> <tr><td>30</td><td>33 0</td><td>81</td><td>34</td></tr> <tr><td>35</td><td>39 1</td><td>88</td><td>36</td></tr> <tr><td>40</td><td>45 4</td><td>95</td><td>37</td></tr> <tr><td>45</td><td>51 9</td><td>101</td><td>38</td></tr> <tr><td>50</td><td>58 4</td><td>108</td><td>39</td></tr> <tr><td>55</td><td>65 1</td><td>114</td><td>40</td></tr> <tr><td>60</td><td>72 0</td><td>120</td><td>41</td></tr> <tr><td>65</td><td>79 1</td><td>125</td><td>42</td></tr> <tr><td>70</td><td>86 4</td><td>131</td><td>43</td></tr> <tr><td>75</td><td>93 9</td><td>136</td><td>44</td></tr> <tr><td>80</td><td>101 4</td><td>142</td><td>45</td></tr> <tr><td>85</td><td>109 1</td><td>147</td><td>46</td></tr> <tr><td>90</td><td>117 0</td><td>152</td><td>47</td></tr> <tr><td>95</td><td>125 1</td><td>158</td><td>48</td></tr> <tr><td>100</td><td>133 4</td><td>163</td><td>49</td></tr> </table>

201. 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 retinance of the air. The diminution of velocity produced by friction is very small, and the retinance of the air is a very inconsiderable

(k) This was also observed by Wolfius, Opera Mathematica, tom. i. p. 802. Schol. iv. considerable 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 ascention is in a vertical line. Wolfius observes, in proof of his remark, that the diminution is occasioned also by the weight of the descending 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.

232. The theory of oblique jets has already been discussed in Prop. IX. art. 161. The two following experiments of Bofut contain all that is necessary to be known in practice. When the height NS of the reservoir AB 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 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 fame, we have \( \sqrt{9} \) feet : \( \sqrt{16} \) feet = 11 feet 3 inches 3 lines : 15 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.

SECT. V. Experiments on the Motion of Water in Conduit Pipes.

203. The experiments of the chevalier de Buat, will be given at great length in the article WATER-Works, for which we have been indebted to the late learned Dr Robison. That the reader, however, may be in possession of every thing valuable on a subject of such public importance, we shall at present give a concise view of the experiments of Couplet and Bofut, and of the practical conclusions which they authorize us to form.

204. 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 Bofut afford much information 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. TABLE XIII.—Containing the Quantities of Water discharged by Conduit Pipes of different lengths and diameters, compared with the Quantities discharged from additional tubes inserted in the same Reservoir.

<table> <tr> <th rowspan="2">Constant altitude of the water in the reservoir above the axis of the tube.</th> <th rowspan="2">Length of the conduit Pipes.</th> <th colspan="2">Quantity of water discharged in a minute by an additional tube.</th> <th colspan="2">Quantity of water discharged by the conduit pipe in a minute.</th> <th colspan="2">Ratio between the quantities of water furnished by the tube and the pipe of 16 lines diameter.</th> <th colspan="2">Quantity of water discharged by an additional tube in a minute.</th> <th colspan="2">Quantity of water discharged by the conduit pipe in a minute.</th> <th colspan="2">Ratio between the quantities of water furnished by the tube and the pipe of 24 lines diameter.</th> </tr> <tr> <th>Tube and pipe 16 lines diam.</th> <th>Cubic Inches.</th> <th>Tube and pipe 24 lines diam.</th> <th>Cubic Inches.</th> <th>Tube and pipe 16 lines diam.</th> <th>Cubic Inches.</th> <th>Tube and pipe 24 lines diam.</th> <th>Cubic Inches.</th> <th>Tube and pipe 16 lines diam.</th> <th>Cubic Inches.</th> <th>Tube and pipe 24 lines diam.</th> <th>Cubic Inches.</th> </tr> <tr> <th>Feet.</th> <th>Feet.</th> <th></th> <th></th> <th></th> <th></th> <th></th> <th></th> <th></th> <th></th> <th></th> <th></th> <th></th> <th></th> </tr> <tr> <td>1</td><td>30</td><td>6330</td><td>2778</td><td>1 to .4389</td><td>14243</td><td>7680</td><td>1 to .5392</td> </tr> <tr> <td>1</td><td>60</td><td>6330</td><td>1957</td><td>1 to .3091</td><td>14243</td><td>5564</td><td>1 to .3906</td> </tr> <tr> <td>1</td><td>90</td><td>6330</td><td>1387</td><td>1 to .2507</td><td>14243</td><td>4534</td><td>1 to .3183</td> </tr> <tr> <td>1</td><td>120</td><td>6330</td><td>1331</td><td>1 to .2134</td><td>14243</td><td>3944</td><td>1 to .2769</td> </tr> <tr> <td>1</td><td>150</td><td>6330</td><td>1178</td><td>1 to .1861</td><td>14243</td><td>3486</td><td>1 to .2448</td> </tr> <tr> <td>1</td><td>180</td><td>6330</td><td>1052</td><td>1 to .1662</td><td>14243</td><td>3119</td><td>1 to .2190</td> </tr> <tr> <td>2</td><td>30</td><td>8939</td><td>4666</td><td>1 to .4548</td><td>20112</td><td>11219</td><td>1 to .5378</td> </tr> <tr> <td>2</td><td>60</td><td>8939</td><td>2888</td><td>1 to .3231</td><td>20112</td><td>8190</td><td>1 to .4072</td> </tr> <tr> <td>2</td><td>90</td><td>8939</td><td>2352</td><td>1 to .2631</td><td>20112</td><td>6812</td><td>1 to .3877</td> </tr> <tr> <td>2</td><td>120</td><td>8939</td><td>2011</td><td>1 to .2250</td><td>20112</td><td>5885</td><td>1 to .2926</td> </tr> <tr> <td>2</td><td>150</td><td>8939</td><td>1762</td><td>1 to .1971</td><td>20112</td><td>5232</td><td>1 to .2601</td> </tr> <tr> <td>2</td><td>180</td><td>8939</td><td>1583</td><td>1 to .1770</td><td>20112</td><td>4710</td><td>1 to .2341</td> </tr> <tr> <td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td> </tr> </table>

Deductions from the preceding table.

205. 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.

Causes of the retardation of water in moving pipes.

206. By examining some of the experiments in the foregoing table, it will appear, that the water sometimes looses \( \frac{9}{10} \)ths 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.

207. When the altitude of the reservoir was two feet, the diminution of discharge, and consequently of velocity, was greater than when the height of the reservoir was only one foot. The cause of this is manifest. Friction increases with the velocity, because a greater number of obstructions are encountered in a certain time, and the velocities are as the square roots of the altitudes; therefore friction must also be as the square roots of the altitudes of the reservoir. On some occasions Coulomb found that the friction of solid bodies diminished with an augmentation of velocity, but there is no ground for supposing that this takes place in the case of fluids.

208. When the pipe is inclined to the horizon, as CGF, the water will move with a greater velocity than in the horizontal tube CG h f. In the former case, the relative gravity of the water, which is to its absolute gravity as F f to C f, 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 CF was to F f 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 5828 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, D m, E n, and F o; AB being the surface of the water in the reservoir. The preceding numbers, representing the quantities discharged at F, E, when the and D, decrease very slowly; consequently by increasing the inclination 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 f CF is about 6° 31', or when F f is the eighth or ninth part of CF. The quantities discharged at C, D, E, and F, will be then equal, and friction will have consumed the velocity arising from the relative gravity of the included water.

209. In order to determine the effects produced by flexures or finuofities in conduit pipes, M. Boffut made the following experiments.

<table> <tr> <th>Altitude of the Water in the Reservoir.</th> <th>Form of the conduit Pipes.—See Figures 8. and 9.</th> <th>Quantities of Water discharged in a Minute.</th> </tr> <tr> <td>Feet. Inches.</td> <td></td> <td>Cubic Inches.</td> </tr> <tr> <td>0 4</td> <td>The rectilineal tube MN placed horizontally,</td> <td>576</td> </tr> <tr> <td>1 0</td> <td>The same tube similarly placed,</td> <td>1050</td> </tr> <tr> <td>0 4</td> <td>The same tube bent into the curvilineal form ABC, fig. 8, each flexure lying flat on a horizontal plane, ABC being a horizontal section,</td> <td>540</td> </tr> <tr> <td>1 0</td> <td>The same tube similarly placed,</td> <td>1030</td> </tr> <tr> <td>0 4</td> <td>The same tube placed as in fig. 9, where ABCD is a vertical section, the parts A, B, C, D rising above a horizontal plane, and the parts a, b, c lying upon it,</td> <td>520</td> </tr> <tr> <td>1 0</td> <td>The same tube similarly placed,</td> <td>1028</td> </tr> </table>

210. 1. The two first experiments of the foregoing table shew, that the quantities 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 shew, that a curvilineal 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, it is demonstrable that 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. 9. Fig. 9. 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.

211. A set of valuable experiments on a large scale were made by M. Couplet upon the motion of water in conduit pipes, and are detailed 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 Boffut in the following table, which gives a distinct view of all that they have done on this subject, and will be of great use to the practical hydraulist. TABLE XV. Containing the results of the Experiments of Couplet and Boffit 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.

<table> <tr> <th rowspan="2">Altitude of the Water in the Reservoir.</th> <th rowspan="2">Length of the Conduit Pipe.</th> <th rowspan="2">Diameter of the Conduit Pipes.</th> <th colspan="2">Nature, Position, and form of the Conduit Pipes.</th> <th rowspan="2">Ratio between the Quantities which would be discharged if the Fluid experienced no resistance in the pipes, and the Quantities actually discharged,—or the Ratio between the initial and the final Velocities of the Fluid.</th> </tr> <tr> <th>Feet.</th> <th>Inch. Lines.</th> </tr> <tr> <td>0</td> <td>4</td> <td>0</td> <td>50</td> <td>12</td> <td>Rectilineal and horizontal pipe made of lead,</td> <td>I to 0.281</td> </tr> <tr> <td>1</td> <td>0</td> <td>0</td> <td>50</td> <td>12</td> <td>The same pipe similarly placed,</td> <td>I to 0.305</td> </tr> <tr> <td>0</td> <td>4</td> <td>0</td> <td>50</td> <td>12</td> <td>The same pipe with several horizontal flexures,</td> <td>I to 0.264</td> </tr> <tr> <td>1</td> <td>0</td> <td>0</td> <td>50</td> <td>12</td> <td>Same pipe,</td> <td>I to 0.291</td> </tr> <tr> <td>0</td> <td>4</td> <td>0</td> <td>50</td> <td>12</td> <td>The same pipe with several vertical flexures,</td> <td>I to 0.254</td> </tr> <tr> <td>1</td> <td>0</td> <td>0</td> <td>50</td> <td>12</td> <td>Same pipe,</td> <td>I to 0.290</td> </tr> <tr> <td>1</td> <td>0</td> <td>0</td> <td>180</td> <td>16</td> <td>Rectilineal and horizontal pipe made of white iron,</td> <td>I to 0.166</td> </tr> <tr> <td>2</td> <td>0</td> <td>0</td> <td>180</td> <td>16</td> <td>Same pipe,</td> <td>I to 0.177</td> </tr> <tr> <td>1</td> <td>0</td> <td>0</td> <td>180</td> <td>24</td> <td>Rectilineal and horizontal pipe made of white iron,</td> <td>I to 0.218</td> </tr> <tr> <td>2</td> <td>0</td> <td>0</td> <td>180</td> <td>24</td> <td>Same pipe,</td> <td>I to 0.234</td> </tr> <tr> <td>20</td> <td>11</td> <td>0</td> <td>177</td> <td>16</td> <td>Rectilineal pipe made of white iron, and inclined so that CF (fig. 7.) is to EF as 21.24 is to 241,</td> <td></td> </tr> <tr> <td>13</td> <td>4</td> <td>8</td> <td>118</td> <td>16</td> <td>Rectilineal pipe made of white iron, and inclined like the last,</td> <td>I to 0.2000</td> </tr> <tr> <td>6</td> <td>8</td> <td>4</td> <td>159</td> <td>16</td> <td>Rectilineal pipe made of white iron, and inclined like the last,</td> <td>I to 0.2500</td> </tr> <tr> <td>0</td> <td>9</td> <td>0</td> <td>1782</td> <td>48</td> <td>Conduit pipe almost entirely of iron, with several flexures both horizontal and vertical,</td> <td>I to 0.354</td> </tr> <tr> <td>1</td> <td>9</td> <td>0</td> <td>1782</td> <td>48</td> <td>Same pipe,</td> <td>I to 0.350</td> </tr> <tr> <td>2</td> <td>7</td> <td>0</td> <td>1782</td> <td>48</td> <td>Same pipe,</td> <td>I to 0.2376</td> </tr> <tr> <td>0</td> <td>3</td> <td>0</td> <td>1710</td> <td>72</td> <td>Conduit pipe almost entirely of iron, with several flexures both horizontal and vertical,</td> <td>I to 0.0387</td> </tr> <tr> <td>0</td> <td>5</td> <td>3</td> <td>1710</td> <td>72</td> <td>Same pipe,</td> <td>I to 0.0809</td> </tr> <tr> <td>0</td> <td>5</td> <td>7</td> <td>7020</td> <td>60</td> <td>Conduit pipe, partly stone and partly lead, with several flexures both horizontal and vertical,</td> <td>I to 0.0878</td> </tr> <tr> <td>0</td> <td>11</td> <td>4</td> <td>7020</td> <td>60</td> <td>Same pipe,</td> <td>I to 0.0432</td> </tr> <tr> <td>1</td> <td>4</td> <td>9</td> <td>7020</td> <td>60</td> <td>Same pipe,</td> <td>I to 0.0476</td> </tr> <tr> <td>1</td> <td>9</td> <td>1</td> <td>7020</td> <td>60</td> <td>Same pipe,</td> <td>I to 0.0513</td> </tr> <tr> <td>2</td> <td>1</td> <td>0</td> <td>7020</td> <td>60</td> <td>Same pipe,</td> <td>I to 0.0532</td> </tr> <tr> <td>12</td> <td>1</td> <td>3</td> <td>3600</td> <td>144</td> <td>Conduit pipe made of iron, with flexures both horizontal and vertical,</td> <td>I to 0.0541</td> </tr> <tr> <td>12</td> <td>1</td> <td>3</td> <td>3600</td> <td>216</td> <td>Conduit pipe made of iron, with several flexures both horizontal and vertical,</td> <td>I to 0.0992</td> </tr> <tr> <td>4</td> <td>7</td> <td>6</td> <td>4740</td> <td>216</td> <td>Conduit pipe made of iron, with several flexures both horizontal and vertical,</td> <td>I to 0.1653</td> </tr> <tr> <td>20</td> <td>3</td> <td>0</td> <td>14040</td> <td>144</td> <td>Conduit pipe made of iron, with several flexures both horizontal and vertical,</td> <td>I to 0.0989</td> </tr> </table>

212. In order to shew 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 Table VI. art. 185, 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; \( \sqrt{7070 : 40000} = 12 \) lines or 1 inch : 28.4 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 40000 cubic inches, its diameter must be increased. The following analogy, therefore, will furnish us with this new diameter; \( \sqrt{5000 : 40000} = 28.54 \) lines : 80.73 lines, or 6 inches

8 1/2 lines, the diameter of the pipe which will discharge 40000 cub. inches of water when its length is 2400 feet.

SECT. VI. Experiments on the Pressure exerted upon Pipes by the water which flows through them.

213. The pressure exerted upon the fides of conduit pipes by the included water, has been already investigated theoretically in Prop. X. 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. This 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, founded on the experiments of Boffut, contains the quantities of water discharged from a lateral orifice about 3 1/2 lines in diameter, according to theory and experiment.

<table> <tr> <th rowspan="2">Altitude of the Water in the Reservoir.</th> <th rowspan="2">Length of the Conduit Pipe.</th> <th colspan="2">Quantities of Water discharged in 1 Minute, according to Theory.</th> <th colspan="2">Quantities of Water discharged in 1 Minute according to Experiment.</th> </tr> <tr> <th>Feet.</th> <th>Cubic Inches</th> <th>Feet.</th> <th>Cubic Inches</th> </tr> <tr><td>1</td><td>30</td><td>176</td><td>171</td></tr> <tr><td>1</td><td>60</td><td>186</td><td>186</td></tr> <tr><td>1</td><td>90</td><td>199</td><td>190</td></tr> <tr><td>1</td><td>120</td><td>191</td><td>191</td></tr> <tr><td>1</td><td>150</td><td>192</td><td>193</td></tr> <tr><td>1</td><td>180</td><td>193</td><td>194</td></tr> <tr><td>2</td><td>30</td><td>244</td><td>240</td></tr> <tr><td>2</td><td>60</td><td>259</td><td>256</td></tr> <tr><td>2</td><td>90</td><td>264</td><td>261</td></tr> <tr><td>2</td><td>120</td><td>267</td><td>264</td></tr> <tr><td>2</td><td>150</td><td>268</td><td>265</td></tr> <tr><td>2</td><td>180</td><td>269</td><td>266</td></tr> </table>

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.

SECT. VII. Experiments on the Motion of Water in Canals.

214. AMONG the numerous experiments which have been made on this important subject, those of the Abbé Boffut seem entitled to the greatest confidence. His experiments 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 5 inches, and its vertical height sometimes half an inch, and at other times an inch. The fides of this orifice were made of copper, and rising perpendicularly from the fide 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 employed in reaching these points of division. The arrival of the water at these points was signified 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.

<table> <tr> <th rowspan="2">Altitude of the water in the reservoir.</th> <th colspan="2">Fl. In.</th> <th colspan="2">Fl. In.</th> <th colspan="2">Fl. In.</th> <th colspan="2">Fl. In.</th> <th colspan="2">Fl. In.</th> <th rowspan="2">Space run through by the water.</th> </tr> <tr> <th>11</th><th>8</th> <th>7</th><th>8</th> <th>3</th><th>8</th> <th>11</th><th>8</th> <th>7</th><th>8</th> <th>3</th><th>8</th> </tr> <tr> <td>Vertical breadth of the orifice.</td> <td>1/2 an inch.</td><td>1/2 an inch.</td> <td>1/2 an inch.</td><td>1/2 an inch.</td> <td>1 inch.</td><td>1 inch.</td> <td>1 inch.</td><td>1 inch.</td> <td>1 inch.</td><td>1 inch.</td> <td>Feet.</td> </tr> <tr> <td>Time in which the number of feet in column seventh are run through by the water.</td> <td>2"</td><td>3"—</td><td>3"+</td><td>2"</td><td>2"+</td><td>3"—</td><td>21</td> <td>5—</td><td>7</td><td>9</td><td>4</td><td>5</td><td>6+</td><td>42</td> <td>10—</td><td>13</td><td>17+</td><td>7</td><td>9</td><td>11+</td><td>63</td> <td>16—</td><td>20</td><td>27+</td><td>11</td><td>14</td><td>18+</td><td>84</td> <td>23+</td><td>28+</td><td>38+</td><td>16½</td><td>20</td><td>26</td><td>105</td> </tr> </table>

215. It appears from column 1ft, 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 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. 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 fines + and — affixed to the numbers in the preceding table indicate, that these numbers are a little too great or too small.

216. The following experiments were made on inclined canals with different declivities, and will be of great use to the practical hydraulist. The inclination of the canal is the vertical distance of one of its extremities from a horizontal line which passes through its canals, other extremity.

<table> <tr> <th colspan="12">Table XVIII. Containing the Velocity of Water in a Rectangular inclined Canal 105 Feet long, and under different Altitudes of Fluid in the Reservoir.</th> </tr> <tr> <th>Altitude of water in the reservoir.</th> <th>Ft.</th> <th>In.</th> <th>Ft.</th> <th>In.</th> <th>Ft.</th> <th>In.</th> <th>Ft.</th> <th>In.</th> <th>Ft.</th> <th>In.</th> <th>Ft.</th> <th>In.</th> <th>Space run through by the Water.</th> </tr> <tr> <td>Inclination of the canal.</td> <td> Ft. In. </td> <td> Ft. In. </td> <td> Ft. In. </td> <td> Ft. In. </td> <td> Ft. In. </td> <td> Ft. In. </td> <td> Ft. In. </td> <td> Ft. In. </td> <td> Ft. In. </td> <td> Ft. In. </td> <td> Ft. In. </td> <td> Feet. </td> </tr> <tr> <td>Height of the orifice</td> <td> 4" 1/2 an inch. </td> <td> 4"+ 14+ 26 </td> <td> 6"+ 18+ 34+ </td> <td> 3'1/2 11 21 </td> <td> 4+ 14 25+ </td> <td> 6 18+ 31+ </td> <td> 35 70 105 </td> </tr> <tr> <td>Inclination of the canal.</td> <td> Ft. In. </td> <td> Ft. In. </td> <td> Ft. In. </td> <td> Ft. In. </td> <td> Ft. In. </td> <td> Ft. In. </td> <td> Ft. In. </td> <td> Ft. In. </td> <td> Ft. In. </td> <td> Ft. In. </td> <td> Ft. In. </td> <td> Feet. </td> </tr> <tr> <td>Height of the orifice</td> <td> 1 inch. </td> <td> 3" 8 15 </td> <td> 4"— 9+ 19— </td> <td> 5"— 13— 23— </td> <td> 3"— 7 14 </td> <td> 4"— 9 16 </td> <td> 5— 12 21 </td> <td> 35 70 105 </td> </tr> <tr> <td>Inclination of the canal.</td> <td> Ft. In. </td> <td> Ft. In. </td> <td> Ft. In. </td> <td> Ft. In. </td> <td> Ft. In. </td> <td> Ft. In. </td> <td> Ft. In. </td> <td> Ft. In. </td> <td> Ft. In. </td> <td> Ft. In. </td> <td> Ft. In. </td> <td> Feet. </td> </tr> <tr> <td>Height of the orifice</td> <td> 1 inch. </td> <td> 2"+ 7 13 </td> <td> 4"— 9 15 </td> <td> 4"— 10 17 1/2 </td> <td> 2"+ 6 12 </td> <td> 3"+ 8 13 </td> <td> 4"— 9 15+ </td> <td> 35 70 105 </td> </tr> <tr> <td>Inclination of the canal.</td> <td> Ft. In. </td> <td> Ft. In. </td> <td> Ft. In. </td> <td> Ft. In. </td> <td> Ft. In. </td> <td> Ft. In. </td> <td> Ft. In. </td> <td> Ft. In. </td> <td> Ft. In. </td> <td> Ft. In. </td> <td> Ft. In. </td> <td> Feet. </td> </tr> <tr> <td>Height of the orifice</td> <td> 1 inch. </td> <td> 2"+ 6 10 </td> <td> 3"— 7+ 12 </td> <td> 4"— 9 14 </td> <td> 2"+ 6 9 </td> <td> 3"— 6 10 </td> <td> 4"— 8 12 </td> <td> 35 70 105 </td> </tr> <tr> <td>Inclination of the canal.</td> <td> Feet. </td> <td> Feet. </td> <td> Feet. </td> <td> Feet. </td> <td> Feet. </td> <td> Feet. </td> <td> Feet. </td> <td> Feet. </td> <td> Feet. </td> <td> Feet. </td> <td> Feet. </td> <td> Feet. </td> <td> Feet. </td> </tr> <tr> <td>In the three first columns the height of the orifice was 1/2 an inch, and in the three last 1 inch.</td> <td> Half sec. 2+ 7 12 17 21+ </td> <td> Half sec. 3+ 8+ 13+ 18+ 23+ </td> <td> Half sec. 4+ 10 16 22 28 </td> <td> Half sec. 2 5 9 13 17 </td> <td> Half sec. 3+ 7 11 15 19 </td> <td> Half sec. 3 8 13 18 22 </td> <td> 21 42 63 84 105 </td> </tr> <tr> <td>Inclination of the canal.</td> <td> Feet. 11 </td> <td> Feet. 11 </td> <td> Feet. 11 </td> <td> Feet. 11 </td> <td> Feet. 11 </td> <td> Feet. 11 </td> <td> Feet. 11 </td> <td> Feet. 11 </td> <td> Feet. 11 </td> <td> Feet. 11 </td> <td> Feet. 11 </td> <td> Feet. 11 </td> <td> Feet. 11 </td> </tr> <tr> <td>Height of the orifice</td> <td> 1 1/2 inches.</td> <td> Half sec. 2 8+ 12 15+ </td> <td> Half sec. 3 10 13 17 </td> <td> Half sec. 3+ 7 11+ 15 </td> <td> 21 42 63 84 </td> <td> 105 </td> </tr> </table>

Time in which the number of feet in the last col. is run through by the water. 217. In the preceding experiments the velocity of the first portion of water that issues from the reservoir was only observed; but when the current is once established, and its velocity permanent, it moves with greater rapidity, and there is always a fixed proportion between the velocity of the first portion of water and the permanent velocity of the established current. The cause of this difference Buffet does not seem to have thoroughly comprehended, when he ascribes it to a diminution of friction when the velocity becomes permanent. The velocity of the first portion of water that issues from the reservoir was measured by its arrival at certain divisions of the canal, consequently the velocity thus determined was the mean velocity of the water. The velocity of the established current, on the contrary, was measured by light bodies floating upon its surface, at the centre of the canal, therefore the velocity thus determined was the superficial velocity of the stream. But the velocity of the superficial central filaments must be the greatest of all, because being at the greatest distance from the sides and bottom of the canal they are less affected by friction than any of the adjacent or inferior filaments, and are not retarded by the weight of any superincumbent fluid. The superficial velocity of the current must of consequence be greater than its mean velocity, or, in other words, the velocity of the established current must exceed the velocity of the first portion of water. The following table contains the experiments of Buffet on this subject; the canal being of the same size as in the former experiments, but 600 feet long, and its inclination one-tenth of the whole, or 59,722 feet.

<table> <tr> <th rowspan="2">Altitude of the water in the reservoir.</th> <th colspan="2">Vertical breadth of the orifice 1 inch.</th> <th colspan="2">Vertical breadth of the orifice 2 inches.</th> <th rowspan="2">Space run through by the water.</th> </tr> <tr> <th>Vel. of the 1st portion of water.</th> <th>Vel. of the established current.</th> <th>Vel. of the 1st portion of water.</th> <th>Vel. of the established current.</th> <th>Feet.</th> </tr> <tr> <td>4 o</td> <td>10</td> <td>8</td> <td>8</td> <td>7</td> <td>100</td> </tr> <tr> <td>4 o</td> <td>20+</td> <td>17</td> <td>17</td> <td>14½</td> <td>200</td> </tr> <tr> <td>4 o</td> <td>31—</td> <td>26</td> <td>26</td> <td>22</td> <td>300</td> </tr> <tr> <td>4 o</td> <td>42—</td> <td>35</td> <td>35—</td> <td>29+</td> <td>400</td> </tr> <tr> <td>4 o</td> <td>52½</td> <td>43+</td> <td>43+</td> <td>37—</td> <td>500</td> </tr> <tr> <td>4 o</td> <td>62+</td> <td>52</td> <td>52—</td> <td>44+</td> <td>600</td> </tr> <tr> <td>2 o</td> <td>11</td> <td>10</td> <td>9</td> <td>8—</td> <td>100</td> </tr> <tr> <td>2 o</td> <td>23</td> <td>20</td> <td>19</td> <td>16</td> <td>200</td> </tr> <tr> <td>2 o</td> <td>35</td> <td>30</td> <td>29</td> <td>24</td> <td>300</td> </tr> <tr> <td>2 o</td> <td>46+</td> <td>40</td> <td>39</td> <td>32</td> <td>400</td> </tr> <tr> <td>2 o</td> <td>58</td> <td>49</td> <td>49</td> <td>40</td> <td>500</td> </tr> <tr> <td>2 o</td> <td>69</td> <td>58</td> <td>58</td> <td>48</td> <td>600</td> </tr> <tr> <td>1 o</td> <td>12+</td> <td>12</td> <td>15</td> <td>13</td> <td>100</td> </tr> <tr> <td>1 o</td> <td>25½</td> <td>23+</td> <td>31</td> <td>26½</td> <td>200</td> </tr> <tr> <td>1 o</td> <td>39</td> <td>33</td> <td>47</td> <td>39½</td> <td>300</td> </tr> <tr> <td>0 6</td> <td>11—</td> <td>9</td> <td>13½</td> <td>11½</td> <td>100</td> </tr> <tr> <td>0 6</td> <td>22</td> <td>18—</td> <td>26½</td> <td>23</td> <td>200</td> </tr> <tr> <td>0 6</td> <td>32</td> <td>27</td> <td>39½</td> <td>33½</td> <td>300</td> </tr> </table>

218. In all the experiments related in this chapter, and in those of the Chevalier Buat, which are given in the article WATER-Works, the temperature of the water employed has never been taken into consideration. That the fluidity of water is increased by heat can scarcely admit of a doubt. Professor Leslie, in his ingenious paper on Capillary Action, has proved by experiment that a jet of warm water will spring much higher than a jet of cold water, and that a typhon which discharges cold water only by drops, will discharge water of a high temperature in a continued stream. A similar fact was observed by the ancients. Plutarch (1) warm water in particular assures us, that the clepsydrae or water clocks went slower in winter than in summer, and he seems to attribute this retardation to a diminution of fluidity. It is therefore obvious, that warm water will issue from an aperture with greater velocity than cold water, and that the quantities of fluid discharged from the same orifice, and under the same pressure, will increase with the temperature of the fluid. Hence we may discover the cause of the great discrepancy between the experiments of different philosophers on the motion of fluids.

(1) Ἐλαυνία γὰρ ἡ Ψυχρὸς το ὑδωρ ποιη βαρο και σφαματωδεις, ας εστιν α ταις κλεψυδραις καταμετωπις, βραδιον γαρ ἐκχυνι χυμων ἡ ἐνεργεις. Aquam enim impellens frigus gravem facit et crassam, quod in clepsydris licet observare: tardius enim trahunt hyeme quam aestate. PLUTARCH, Quest. Natural.

On the Resistance of Fluids.

Experiments for determining the effects of heat on the motion of fluids.

219. The writer of this article has a set of experiments in view, by which he expects to determine the precise effects of heat upon the motion of fluids, and to furnish the practical hydraulicist with a more correct formula than that of the Chevalier Buat, for finding, under any given circumstances, the velocity of water and the quantities discharged. He hopes also to be able to determine whether or not the friction of water in conduit pipes varies, as in the case of solid bodies, with the nature of the substances of which the pipes are formed; and to ascertain the effects of different unguents in diminishing the resistance of friction. The result of these experiments will probably be communicated in a subsequent article of this work.

CHAP. III. On the Resistance of Fluids.

Reference to the article Resistance of Fluids.

220. In the article RESISTANCE of Fluids, the reader will find that important subject treated at great length, and with great ability, by the late learned Dr Robison. The researches of preceding philosophers are there given in full detail; their different theories are compared with experiments, and the defects of these minutely considered. Since that article was composed, this intricate subject has been investigated by other writers, and though they have not enriched the science of hydraulics with a legitimate theory of the resistance of fluids, the results of their labours cannot fail to be interesting to every philosopher.

Researches of Coulomb.

221. 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. His experiments are new, and were performed with the greatest accuracy; and the results which he obtained were perfectly conformable to the deductions of theory. We shall therefore endeavour to give the reader some idea of the discoveries which he has made.

222. 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.

* Principia lib. ii. prop. resistance which the air opposed to the oscillatory motion of a globe in small oscillations, he employed a formula of three terms, one of them being as the square of the velocity, the second the \( \frac{3}{4} \) power of the velocity, and the third as the simple velocity; and in another part of the work he reduces the formula to two terms, one of which is as the square of the velocity, and the other constant. D. Bernouilli (Comment Petropol. tom. iii. Bernouilli, and v.) also supposes the resistance to be represented by two terms, one as the square of the velocity, and the other constant. M. Gravelende (Elements of Nat. Phil. art. 1911), has found that the pressure of a fluid in motion against a body at rest, is partly proportional to the simple velocity, and partly to the square of the velocity. But when the body moves in a fluid at rest, he found (art. 1975) the resistance proportional to the square of the velocity, and to a constant quantity.—When the body in motion, therefore, meets the fluid at rest, these three philosophers have agreed, 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, incontrovertibly 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 so very small as to escape detection.

224. In order to apply the principle of torsion to the resistance of fluids, M. Coulomb made use of the apparatus represented in fig. 1. On the horizontal arm LK, which may be supported by a vertical stand, is fixed the small circle f e, perforated in the centre, so as to admit the cylindrical pin b a. Into a slit in the extremity of this pin is fastened, by means of a forew, the brafs wire a g, 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 g d, whose diameter is about four-tenths of an inch. The cylinder g d is perpendicular to the disc DS, whose circumference is divided into 480 equal parts. When this horizontal disc is at rest, which happens when the torsion of the brafs wire is nothing, the index RS is placed upon the point o, the zero of the circular scale. The small rule R m may be elevated or depressed at pleasure round its axis n, and the stand GH which supports it may be brought into any position round the horizontal disc. The lower extremity of the cylinder g d 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 brafs wire. In order to produce these oscillations, the disc 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 disc is then left to itself; the force of torsion causes it to oscillate, and the successive diminutions of these oscillations are carefully observed. A simple formula gives in weights the force of torsion that produces the oscillations; and another formula well known to geometers, determines (by an approximation sufficiently accurate in practice), by means of the successive diminution of the oscillations compared with their amplitude, what is the law of the resistance, relative to the velocity, which produces these diminutions. 225. 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 refinance 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 formulae 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 refinance at the point of floatation than at any other part. This refinance, 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.

226. These and other inconveniences which might be mentioned, are so inseparable from the use of the pendulum, that Newton and Bernouilli have not been able to determine the laws of the refinance of fluids in very slow motions. When the refinance 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 disc. 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 disc, 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 refinance 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.

227. In the first set of experiments made by Coulomb, he attached to the lower extremity of the cylinder g d a circular plate of white iron, about 195 millimetres in diameter, and made it move so slowly, that the part of the refinance proportional to the square of the velocity, wholly disappeared. For if, in any particular case, the portion of the refinance 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 1/100th 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.

228. When the oscillations of the white iron plate were to flow, that the part of the refinance 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 refinance which diminished the oscillations of the horizontal plate was uniformly proportional to the simple velocity, and that the other part of the refinance, which follows the ratio of the square of the velocity, produced no sensible change upon the motion of the white iron disc.—He found also, in conformity with theory, that the moments of refinance in different circular plates moving their centre in a fluid, are as the fourth power of the diameters of these circles; and that, when a circle of 195 millimetres (6.677 English inches) in diameter, moved round its centre in water, so that its circumference had a velocity of 140 millimetres (5.512 English inches) per second, the momentum of refinance 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 millimetres (5.63 English inches) in length.

229. 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 moments of the refinance of different circles, 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 17.5 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.

230. In order to ascertain whether or not the refinance of a body moving in a fluid was influenced by the nature of its surface, M. Coulomb anointed the surface once not influenced by the white iron plate with tallow, and wiped it part. ly 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 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.

231. 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 (.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 harpichord wire, numbered 7 in commerce, and suspended to it a cylinder of copper, like \( g_d \), fig. 1. 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.

232. The next object of M. Coulomb was to ascertain the resistance experienced by cylinders that moved very slowly, and perpendicular to their axes; 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 therefore 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 millimetres (.9803 English inches) long. The first cylinder was 0.87 millimetres (0.0342 English inches or \( \frac{1}{35} \) of an inch) in circumference, the second 11.2 millimetres (0.4409 English inches), and the third 21.1 millimetres (.88307 English inches). They were fixed by their middle under the cylindrical piece \( dg \), so as to form two horizontal radii, whose length was 124.5 millimetres (.4901 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.

233. In order to explain these results, M. Coulomb cause of very justly supposes, that in consequence of the mutual this, 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 augmented increased, to be 1.5 millimetres (\( \frac{59}{1000} \) of an English b., \( \frac{59}{1000} \) inch) so that the diameter of the augmented cylinder will exceed the diameter of the real cylinder by double that quantity, or \( \frac{118}{1000} \) of an inch.

234. The part of the resistance varying with the square of the velocity, or that arising from the inertia due to 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 \( \frac{1}{1000} \) 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; radii are 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 this difference 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 \( r \).

235. In determining experimentally the part of the momentum of resistance proportional to the velocity, by two cylinders of the same diameter, but of different diameters, M. Coulomb found that this momentum was proportional 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.

236. When the cylinder 0.9803 inches in length, and 0.04429 inches in circumference, was made to oscillate in the fluid with a velocity of 5.17 inches per second, the part of the resistance r was equal to 58 milligrammes, or .8932 troy grains. And when the velocity was 0.3937 inches per second, the resistance r was 0.00414 grammes, or 0.637 troy grains.

237. 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.

238. In a subsequent memoir Coulomb proposes to determine numerically the part of the resistance proportional to the square of the velocity, and to ascertain the resistance of globes with plain, convex, and concave surfaces. He has found in general that the resistance of bodies not entirely immersed in the fluid is much fluids, greater than that of bodies which are wholly immerfed; and he promises to make farther experiments upon this point. We intended on the present occasion to have given the reader a more complete view of the researches of this ingenious philosopher; but these could not well be understood without a knowledge of his investigations respecting the force of torsion, which we have not yet had an opportunity of communicating. In the article MECHANICS, however, we shall introduce the reader to this interesting subject; and may afterwards have an opportunity of making him farther acquainted with those researches of Coulomb, of which we have at present given only a general view.

239. The subject of the resistance of fluids has been recently treated by the learned Dr Hutton of Woolwich, of Dr Hut- ton. 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. The following table contains the results of many interesting experiments. The numbers in the 9th column represent the exponents of the power of the velocity which the resistances in the 8th column bear to each other.

<table> <tr> <th rowspan="2">Velocity per se- cond.</th> <th colspan="2">Small hemisphere, 4 3/8 inches dia. flat side.</th> <th colspan="2">Large hemisphere 6 5/8 inches diameter.</th> <th colspan="2">Cone 6 1/2 inches diameter.</th> <th colspan="2">Cylinder 6 5/8 inches diameter.</th> <th colspan="2">Globe 6 5/8 inches diameter.</th> <th rowspan="2">Power of the vel. to which the resistance is proportional.</th> </tr> <tr> <th>Flat side.</th> <th>Round side.</th> <th>Vertex.</th> <th>Bafe.</th> <th>Ounces av.</th> <th>Ounces av.</th> <th>Ounces av.</th> <th>Ounces av.</th> <th>Ounces av.</th> <th>Ounces av.</th> </tr> <tr><td>Feet.</td><td>Ounces av.</td><td>Ounces av.</td><td>Ounces av.</td><td>Ounces av.</td><td>Ounces av.</td><td>Ounces av.</td><td>Ounces av.</td><td>Ounces av.</td><td>Ounces av.</td><td>Ounces av.</td><td>Ounces av.</td><td></td></tr> <tr><td>3</td><td>.028</td><td>.051</td><td>.020</td><td>.028</td><td>.064</td><td>.050</td><td>.027</td></tr> <tr><td>4</td><td>.048</td><td>.096</td><td>.039</td><td>.048</td><td>.109</td><td>.090</td><td>.047</td></tr> <tr><td>5</td><td>.072</td><td>.148</td><td>.063</td><td>.071</td><td>.162</td><td>.143</td><td>.068</td></tr> <tr><td>6</td><td>.103</td><td>.211</td><td>.092</td><td>.098</td><td>.225</td><td>.205</td><td>.094</td></tr> <tr><td>7</td><td>.141</td><td>.284</td><td>.123</td><td>.129</td><td>.298</td><td>.278</td><td>.125</td></tr> <tr><td>8</td><td>.184</td><td>.368</td><td>.160</td><td>.168</td><td>.382</td><td>.360</td><td>.162</td></tr> <tr><td>9</td><td>.233</td><td>.464</td><td>.199</td><td>.211</td><td>.478</td><td>.456</td><td>.205</td></tr> <tr><td>10</td><td>.287</td><td>.573</td><td>.242</td><td>.260</td><td>.587</td><td>.565</td><td>.255</td></tr> <tr><td>11</td><td>.349</td><td>.698</td><td>.292</td><td>.315</td><td>.712</td><td>.688</td><td>.310</td><td>2.052</td></tr> <tr><td>12</td><td>.418</td><td>.836</td><td>.347</td><td>.376</td><td>.850</td><td>.826</td><td>.370</td><td>2.042</td></tr> <tr><td>13</td><td>.492</td><td>.988</td><td>.409</td><td>.440</td><td>1.000</td><td>.979</td><td>.435</td><td>2.036</td></tr> <tr><td>14</td><td>.573</td><td>1.154</td><td>.478</td><td>.512</td><td>1.166</td><td>1.145</td><td>.505</td><td>2.031</td></tr> <tr><td>15</td><td>.661</td><td>1.336</td><td>.552</td><td>.589</td><td>1.346</td><td>1.327</td><td>.581</td><td>2.031</td></tr> <tr><td>16</td><td>.754</td><td>1.538</td><td>.634</td><td>.673</td><td>1.546</td><td>1.526</td><td>.663</td><td>2.033</td></tr> <tr><td>17</td><td>.853</td><td>1.757</td><td>.722</td><td>.762</td><td>1.703</td><td>1.745</td><td>.752</td><td>2.038</td></tr> <tr><td>18</td><td>.959</td><td>1.988</td><td>.818</td><td>.858</td><td>2.002</td><td>1.986</td><td>.848</td><td>2.044</td></tr> <tr><td>19</td><td>1.073</td><td>2.998</td><td>.921</td><td>.959</td><td>2.260</td><td>2.246</td><td>.949</td><td>2.047</td></tr> <tr><td>20</td><td>1.196</td><td>2.542</td><td>1.033</td><td>1.069</td><td>2.540</td><td>2.528</td><td>1.057</td><td>2.051</td></tr> <tr><td colspan="2">Mean proportional numbers.</td><td>140</td><td>288</td><td>119</td><td>126</td><td>291</td><td>285</td><td>124</td><td>2.040</td></tr> <tr><td colspan="2">1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td><td>9</td></tr> </table> 240. From the preceding experiments we may draw the following conclusions: 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 from the 9th column that the exponent increases with the velocity. 3. The round and sharp ends of folds 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 2 1/2 to 1, instead of 2 to 1, as given by theory. 5. The resistance on the base of the cone is to the resistance on the vertex nearly as 2 1/2 to 1; and in the same ratio is radius to the fine 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.

241. The following table contains the resistance sustained by a globe 1.965 inches in diameter. The fourth column is the quotient of the resistance by experiment, divided by the theoretical resistance.

<table> <tr> <th>Velocity of the Globe per second</th> <th>Resistance by experiment.</th> <th>Resistance by theory.</th> <th>Ratio between the experimental and theoretical resistances.</th> <th>Power of the velocity to which the resistance is proportional.</th> </tr> <tr> <td>Feet.</td> <td>Oz. avoir.</td> <td>Oz. avoir.</td> <td></td> <td></td> </tr> <tr><td>5</td><td>0.006</td><td>0.005</td><td>1.20</td><td></td></tr> <tr><td>10</td><td>0.0245</td><td>0.020</td><td>1.23</td><td></td></tr> <tr><td>15</td><td>0.055</td><td>0.044</td><td>1.25</td><td></td></tr> <tr><td>20</td><td>0.100</td><td>0.079</td><td>1.27</td><td></td></tr> <tr><td>25</td><td>0.157</td><td>0.123</td><td>1.28</td><td>2.022</td></tr> <tr><td>30</td><td>0.23</td><td>0.177</td><td>1.30</td><td>2.039</td></tr> <tr><td>40</td><td>0.42</td><td>0.314</td><td>1.33</td><td>2.068</td></tr> <tr><td>50</td><td>0.67</td><td>0.491</td><td>1.36</td><td>2.075</td></tr> <tr><td>100</td><td>2.72</td><td>1.964</td><td>1.38</td><td>2.059</td></tr> <tr><td>200</td><td>11</td><td>7.9</td><td>1.40</td><td>2.041</td></tr> <tr><td>300</td><td>25</td><td>18.7</td><td>1.41</td><td>2.039</td></tr> <tr><td>400</td><td>43</td><td>31.4</td><td>1.43</td><td>2.039</td></tr> <tr><td>500</td><td>72</td><td>49</td><td>1.47</td><td>2.044</td></tr> <tr><td>600</td><td>107</td><td>71</td><td>1.51</td><td>2.051</td></tr> <tr><td>700</td><td>151</td><td>96</td><td>1.57</td><td>2.059</td></tr> <tr><td>800</td><td>205</td><td>126</td><td>1.63</td><td>2.067</td></tr> <tr><td>900</td><td>271</td><td>159</td><td>1.70</td><td>2.077</td></tr> <tr><td>1000</td><td>350</td><td>196</td><td>1.78</td><td>2.086</td></tr> <tr><td>1100</td><td>442</td><td>238</td><td>1.86</td><td>2.095</td></tr> <tr><td>1200</td><td>546</td><td>283</td><td>1.90</td><td>2.102</td></tr> <tr><td>1300</td><td>661</td><td>332</td><td>1.99</td><td>2.107</td></tr> <tr><td>1400</td><td>785</td><td>385</td><td>2.04</td><td>2.111</td></tr> <tr><td>1500</td><td>916</td><td>442</td><td>2.07</td><td>2.113</td></tr> <tr><td>1600</td><td>1051</td><td>503</td><td>2.09</td><td>2.113</td></tr> <tr><td>1700</td><td>1186</td><td>568</td><td>2.08</td><td>2.111</td></tr> <tr><td>1800</td><td>1319</td><td>636</td><td>2.07</td><td>2.108</td></tr> <tr><td>1900</td><td>1447</td><td>709</td><td>2.04</td><td>2.104</td></tr> <tr><td>2000</td><td>1569</td><td>786</td><td>2.00</td><td>2.098</td></tr> </table>

242. It appears from a comparison of the 2d, 3d, and 4th columns, that when the velocity is small the resistance by experiment is nearly equal to that deduced from theory; but that as the velocity increases, the former gradually exceeds the latter till the velocity is 1300 feet per second, when it becomes twice as great. The difference between the two resistances then increases, and reaches its maximum between the velocities of 1600 and 1700 feet. It afterwards decreases gradually as the velocity increases, and at the velocity of 2000 the resistance by experiment is again double of the theoretical resistance.—By considering the numbers in column 5th it will be seen, that in flow motions the resistances are nearly as the squares of the velocities; that this ratio increases gradually, though not regularly, till at the velocity of 1500 or 1600 feet it arrives at its maximum. It then gradually diminishes as the velocity increases.

Conclusions similar to these were deduced from experiments made with globes of a larger size.

243. The following table contains the resistance of a plane inclined at various angles, according to experiment, and according to a formula deduced from the experiments.

<table> <tr> <th>Inclination of the plane.</th> <th>Resistances by experiment.</th> <th>Resistances by the formula .84r^1.84c.</th> <th>Sines of the angles to radius .840.</th> </tr> <tr> <td>Degrees.</td> <td>Oz. avoir.</td> <td>Oz. avoir.</td> <td></td> </tr> <tr><td>0</td><td>.000</td><td>.000</td><td>.000</td></tr> <tr><td>5</td><td>.015</td><td>.009</td><td>.073</td></tr> <tr><td>10</td><td>.044</td><td>.035</td><td>.146</td></tr> <tr><td>15</td><td>.082</td><td>.076</td><td>.217</td></tr> <tr><td>20</td><td>.133</td><td>.131</td><td>.287</td></tr> <tr><td>25</td><td>.200</td><td>.199</td><td>.355</td></tr> <tr><td>30</td><td>.278</td><td>.278</td><td>.420</td></tr> <tr><td>35</td><td>.362</td><td>.363</td><td>.482</td></tr> <tr><td>40</td><td>.448</td><td>.450</td><td>.540</td></tr> <tr><td>45</td><td>.534</td><td>.535</td><td>.594</td></tr> <tr><td>50</td><td>.619</td><td>.613</td><td>.643</td></tr> <tr><td>55</td><td>.684</td><td>.685</td><td>.688</td></tr> <tr><td>60</td><td>.729</td><td>.736</td><td>.727</td></tr> <tr><td>65</td><td>.770</td><td>.778</td><td>.761</td></tr> <tr><td>70</td><td>.823</td><td>.826</td><td>.789</td></tr> <tr><td>75</td><td>.835</td><td>.836</td><td>.811</td></tr> <tr><td>80</td><td>.839</td><td>.839</td><td>.827</td></tr> <tr><td>85</td><td>.840</td><td>.840</td><td>.838</td></tr> <tr><td>90</td><td>.840</td><td>.840</td><td>.842</td></tr> <tr><td>1</td><td>2</td><td>3</td><td>4</td></tr> </table>

244. The plane with which the preceding experiments were performed was 32 square inches, and always moved with a velocity of 12 feet per second. The resistances which this plane experienced are contained in column 2d. From the numbers in that column Dr Hutton deduced the formula .84r^1.84c, where r is the fine, and c the cosine of the angles of inclination in the first column. The resistances computed from this formula are contained in column 3d, and agree very near- ly with the resistances deduced from experiment. The 4th column contains the fines of the angles in the first column to a radius .84, in order to compare them with the resistances which have obviously no relation either to the fines of the angles or to any power of the fines. From the angle of 0° to about 65° the resistances are less than the fines; but from 65° to 90° they are somewhat greater.

245. 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 c.66 feet in a second. The angles at which the planes struck the fluid are contained in the first column. The second column shews 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 shews the power of the fine of the angle to which the resistance is proportional, and was computed in the following manner. Let o be the fine of the angle, radius being 1, and r the resistance at that angle. Suppose r to vary as \( s^m \), then we have \( r^m : s^m = 0.2321 : r \); hence \( s^m = \frac{r}{0.2321} \), and therefore \( m = \log r - \log 0.2321 \), and by substituting their corresponding values, instead of r and s we shall have the values of m or the numbers in the fourth column.

<table> <tr> <th>Angle of inclination.</th> <th>Resistance by experiment.</th> <th>Resistance by theory.</th> <th>Power of the fine of the angle to which the resistance is proportionate.</th> </tr> <tr> <th>Degrees.</th> <th>Troy ounces.</th> <th>Troy ounces.</th> <th>Exponents.</th> </tr> <tr> <td>10</td> <td>0.0112</td> <td>0.0012</td> <td>1.73</td> </tr> <tr> <td>20</td> <td>0.0364</td> <td>0.0093</td> <td>1.73</td> </tr> <tr> <td>30</td> <td>0.0769</td> <td>0.0290</td> <td>1.54</td> </tr> <tr> <td>40</td> <td>0.1174</td> <td>0.0616</td> <td>1.54</td> </tr> <tr> <td>50</td> <td>0.1552</td> <td>0.1043</td> <td>1.51</td> </tr> <tr> <td>60</td> <td>0.1902</td> <td>0.1476</td> <td>1.38</td> </tr> <tr> <td>70</td> <td>0.2125</td> <td>0.1926</td> <td>1.42</td> </tr> <tr> <td>80</td> <td>0.2237</td> <td>0.2217</td> <td>2.41</td> </tr> <tr> <td>90</td> <td>0.2321</td> <td>0.2321</td> <td></td> </tr> <tr> <td>1</td> <td>2</td> <td>3</td> <td>4</td> </tr> </table>

246. According to the theory the resistance should vary as the cube of the fine, 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 revolution acts parallel to the plane, but which according to experiments is really a part of the force which acts upon the plane.

247. Mr Vince made also a number of experiments on the resistance of hemispheres, globes, and cylinders, which moved with a velocity of c.542 feet per second, hemi-spheres, and hemisphere was to the resistance on its base as 0.034 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.

248. The following results were obtained, when the plane was struck by the moving fluid. The 2d column of the following table contains the resistance by experiment, and the 3d column the resistance by theory from the perpendicular force, supposing it to vary as the sine of the inclination.

<table> <tr> <th>Angle of inclination.</th> <th>Resistance by experiment.</th> <th>Resistance by theory.</th> </tr> <tr> <th>Degrees.</th> <th>Oz. dwts. grs.</th> <th>Oz. dwts. grs.</th> </tr> <tr> <td>90</td> <td>1 17 12</td> <td>1 17 12</td> </tr> <tr> <td>80</td> <td>1 17 0</td> <td>1 16 22</td> </tr> <tr> <td>70</td> <td>1 15 12</td> <td>1 15 6</td> </tr> <tr> <td>60</td> <td>1 12 12</td> <td>1 12 11</td> </tr> <tr> <td>50</td> <td>1 18 10</td> <td>1 18 17</td> </tr> <tr> <td>40</td> <td>1 4 10</td> <td>1 4 2</td> </tr> <tr> <td>30</td> <td>0 18 18</td> <td>0 18 18</td> </tr> <tr> <td>20</td> <td>0 12 12</td> <td>0 12 19</td> </tr> <tr> <td>10</td> <td>0 6 4</td> <td>0 6 12</td> </tr> <tr> <td>1</td> <td>2</td> <td>3</td> </tr> </table>

249. It appears from the preceding results, that the resistance varies as the fine of the angle at which the fluid strikes the plane, the difference between theory and experiment being such as might be expected from the necessary inaccuracy of the experiments.

By comparing the preceding table with Table IV. it will be found that the resistance of a plane moving in a fluid is to the resistance of the same plane when struck by the fluid in motion as 5 to 6. 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.

CHAP. IV. On the Oscillation of Fluids, and the Undulation of Waves.

PROP. I.

250. The oscillations of water in a syphon, consisting of two vertical branches and a horizontal one, are isochronous, and have the same duration.

Into the tube MNOP, 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 syphon oscillate round the point y, the water will rise and fall alternately in the vertical branches after the syphon 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 be a pendulum, whose length is equal to half the length of the oscillating column AOPD, and which describes to the lowest point S arches PS, equal to AE; then 2 AE : 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 arch 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.

251. 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.

SCHOLIUM.

252. This subject has been treated in a general manner, by Newton and different philosophers, who have shewn how to determine the time of an oscillation, whatever be the form of the syphon. See the Principia, lib. ii. Prop. 45, 46. Boffut's Traité d'Hydrodynamique, tom. i. Notes sur le Chap. II. Part II. Bernoulli Opera, tom. iii. p. 125, and Encyclopédie, art. Ondes.

PROP. II.

253. 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, the undulations are performed in such a manner, that the highest parts A, C, E 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 syphon. 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 cavities, which is called the breadth of the wave. Now if a pendulum, whose length is one-half BM, performs two oscillations in the above time, it will require a pendulum four times that length to perform only one oscillation in the same time, that is, a pendulum whose length is AC or BD, since \( 4 \times \frac{1}{2} BM = 2 BM = AC \) or \( BD \). Q. E. D.

SCHOLIUM.

254. The explanation of the oscillation of waves contained in the two preceding propositions, was first given by Sir Isaac Newton, in his Principia, lib. ii. Prop. 44. He considered it only as an approximation to the truth, since it supposes the waves to rise and fall perpendicularly like the water in the vertical branches of the syphon, while their real motion is partly circular. The theory of Newton was, nevertheless, adopted by succeeding philosophers, and gave rise to many analogous disquisitions respecting the modulation of waves. Very lately, however, an attempt has been made by M. Flaugergues, to overturn the theory of Newton. From a number of experiments on the motion and figure of waves, an account of which may be seen in the Journal des Scavans, for October 1789, M. Flaugergues concludes, that a wave is not the result of a motion in the particles of water, by which they ascend and descend alternately in a serpentine line, when moving from the place where the water received the shock; but that it is an intumescence which this shock occasions around the place where it is received, by the depression that is there produced. This intumescence afterwards propagates itself circularly, while it removes from the place where the shock first raised it above the level of the stagnant water. A portion of the stagnant water then flows from all sides into the hollow formed at the place where the shock was received; this hollow is thus heaped with fluid, and the water is elevated so as to produce all around another intumescence, or a new wave, which propagates itself circularly as before. The repetition of this effect produces on the surface of the water a number of concentric rings, successively elevated and depressed, which have the appearance of an undulatory motion. This interesting subject has also been discussed by M. La Grange, in his Méchanique Analytique, to which we must refer the reader for farther information. See also some excellent remarks on this subject, in Mr Leslie's Essay on Heat, p. 225, and note 29. PART III. ON HYDRAULIC MACHINERY.

255. 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.

CHAP. I. On Water Wheels.

256. WATER-wheels are divided into three kinds, overhot-wheels, breast-wheels, and underhot-wheels, which derive their names from the manner in which the water is delivered upon their circumferences.

SECT. I. On Overhot-Wheels.

257. An overhot-wheel is a wheel driven by the weight of water, conveyed into buckets disposed on its circumference. It is represented in fig. 5, 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 A a. The equilibrium of the wheel is therefore destroyed; and the power of the bucket A a, 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 O m, c being the centre of gravity of the bucket, and O m a perpendicular let fall from the fulcrum O to the direction c m, in which the force is exerted. In consequence of this destruction of equilibrium, the wheel will move round in the direction AB, the bucket A a will be at d, and the empty bucket b will take the place of A a, 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 n O, and the water in the bucket A a acting with a lever m O. 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 c, where their power, represented by e O, 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 B it is completely discharged. When the wheel begins to move, its velocity will increase rapidly till the quadrant of buckets b e is completely filled. While these buckets are descending through the inferior quadrant e P, 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 A c has reached the point B where it is emptied, the whole circumference nearly 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.

258. In order to find the power of the loaded arch Method of to turn the wheel, or, which is the same thing, to find computing a weight which suspended at the opposite extremity C the moment will balance the loaded arch or keep it in equilibrium, the water we must multiply the weight of water in each bucket in the load, by the length of the virtual lever by which it acts, ed arch, and take the sum of all these momenta for the momentum of the loaded arch. It will be much easier, however, and the result will be the same, if we multiply the weight of all the water on the arch AB, by the distance of its centre of gravity G, from the fulcrum or centre of motion O. Now, by the property of the centre of gravity (see MECHANICS), the distance of the centre of gravity of a circular arch from its centre, is a fourth proportional to half the arch, the radius, and the fine of half the arch. Since the vertical bucket b has no power to turn the wheel if it were filled, and since two or three buckets between B and P are always empty, we may safely suppose that the loaded arch never exceeds 160°, so that if R = radius of the wheel in feet, we shall have the length of half the loaded arch, or 80° = \( 2 \times 3.1416 \times \frac{R}{2} = R \times 1.396 \); and the distance of the centre of gravity from the fulcrum O, \( = GO = R \times \sin 80^\circ \). Now, if N be the number of buckets in the wheel, \( \frac{160}{360} N \) or \( \frac{4N}{9} \), will be the number of buckets in the loaded arch; and if G be the number of ale gallons contained in each bucket, the weight of the water in each bucket will be \( 1.02 \times G \) pounds. avordupois. The weight of the water, therefore, in the loaded arch, will be \( \frac{4N}{9} \times 10.2G \), and consequently the momentum of the loaded arch will be \( \frac{4N}{9} \times 10.2G \times \frac{R \times \sin 80^\circ}{R \times 1.396} = \frac{4N}{9} \times 10.2G \times 0.6338 \) \( = \frac{4N}{9} \times 6.465 \) G pounds avordupois. Hence, we have the following rule: Multiply the constant number 6.465 by \( \frac{4}{9} \) of the number of buckets in the wheel, and this product by the number of ale gallons in each bucket; and the result will be the effective weight, or momentum of the water in the loaded arch. For a description of the best form that can be given to the buckets, see the article WATER-Works. Dr Robison has there recommended a mode of constructing the buckets invented by Mr Burns, who divided each bucket into two by means of a partition; but the writer of this article is assured, on the authority of an ingenious millwright, who wrought with Mr Burns at the time when wheels of this kind were constructed, that the inner bucket is never filled with water, and that much of the power is thus lost. The partition prevents the introduction introduction of the fluid, and the water is driven backwards by the escape of the included air.

259. In the construction of overhot-wheels, it is of great importance to determine what should be the diameter of the wheel relatively to the height of the fall. It is evident that its diameter cannot exceed the height of the fall. Some mechanical writers have demonstrated that, in theory, an overhot-wheel will produce a maximum effect when its diameter is two-thirds of that height, the water being supposed to fall into the buckets with the velocity of the wheel. But this rule is palpably erroneous, and directly repugnant to the results of experiment. For if the height of the fall be 48 feet, the diameter of the wheel will, according to this rule, be 32 feet; and the water having to fall through 16 feet before it reaches the buckets, will have a velocity of 32 feet per second, which, according to the hypothesis, must also be the velocity of the wheel's circumference. But Smeaton has proved, that a maximum effect is produced by an overhot-wheel of any diameter, when its velocity is only three feet per second. The chevalier de Borda has shewn, that overhot-wheels will produce a maximum effect when their diameter is equal to the height of the fall; and this is completely confirmed by Mr Smeaton's experiments. From a great number of trials, Mr Smeaton has concluded, "that the higher the wheel is in proportion to the whole descent, the greater will be the effect." Nor is it difficult to assign the reason of this. The water which is conveyed into the buckets can produce very little effect by its impulse, even if its velocity be great; both on account of the obliquity with which it strikes the buckets, and in consequence of the loss of water occasioned by a considerable quantity of the fluid being dashed over their sides. Instead, therefore, of expecting an increase of effect from the impulse of the water occasioned by its fall through one-third of the whole height, we should allow it to act through this height by its gravity, and therefore make the diameter of the wheel as great as possible. But a disadvantage attends even this rule; for if the water is conveyed into the buckets without any velocity, which must be the case when the diameter of the wheel equals the height of the fall, the velocity of the wheel will be retarded by the impulse of the buckets against the water, and much power would be lost by the water dashing over them. In order, therefore, to avoid all inconveniences, the distance of the spout from the receiving bucket should, in general, be about two or three inches, that the water may be delivered with a velocity a little greater than that of the wheel; or, in other words, the diameter of an overhot-wheel should be two or three inches less than the greatest height of the fall; and yet it is no uncommon thing to see the diameters of these wheels scarcely one-half of that height. In such a construction the loss of power is prodigious.

On the proper velocity of overhot-wheels. Experiments of Deparcieux on the velocity of overhot-wheels.

260. The proper velocity of overhot-wheels is a subject on which mechanical writers have entertained different sentiments. While some have maintained that there is a certain velocity which produces a maximum effect, Deparcieux has endeavoured to prove by a set of ingenious experiments that most work is performed by an overhot-wheel when it moves slowly, and that the more its motion is retarded by increasing the work to be performed, the greater will be the performance of the wheel. In these experiments he employed a small overhot-wheel, 20 inches in diameter, having its circumference furnished with 48 buckets. On the centre or axle of this wheel were placed 4 cylinders of different diameters, the first being 1 inch in diameter, the second 2 inches, the third 3 inches, and the fourth 4 inches. When the experiments are made, a cord is attached to one of the cylinders, and after passing over a pulley a weight is suspended at its other extremity. By moving the wheel upon its axis, the cord winds round the cylinder and raises the weight. In order to diminish the friction, the gudgeons of the wheel are supported by two friction rollers, and before the wheel, a little higher than its axis, is placed a small table which supports a vessel filled with water, having an orifice in the side next the wheel. Above this vessel is placed a large bottle full of water and inverted, having its mouth immersed a few lines in the water, so that it empties itself in proportion as the water in the vessel is discharged from the orifice. The quantity of water thus discharged is always the same, and is conveyed from the orifice by means of a canal to the buckets of the wheel. With this apparatus he obtained the following results.

<table> <tr> <th rowspan="2">Diameters of the Cylinders</th> <th colspan="2">Altitude through which 12 ounces were elevated.</th> <th colspan="2">Altitude through which 24 ounces were elevated.</th> </tr> <tr> <th>Inches.</th> <th>Lines.</th> <th>Inches.</th> <th>Lines.</th> </tr> <tr> <td>1</td> <td>69</td> <td>9</td> <td>49</td> <td>0</td> </tr> <tr> <td>2</td> <td>80</td> <td>6</td> <td>43</td> <td>6</td> </tr> <tr> <td>3</td> <td>85</td> <td>6</td> <td>44</td> <td>6</td> </tr> <tr> <td>4</td> <td>87</td> <td>9</td> <td>45</td> <td>3</td> </tr> </table>

261. When the large cylinders were used, the velocity of the wheel was smaller, because the resistances are proportional to their diameter, the weight being the same. Hence, it appears, by comparing the four results of column 2d with one another, and also the four previous results in column 3d, that when the wheel turns more slowly, the effect, which is in this case measured by the amount of elevation of the weight, always increases. When the weight of 24 ounces was used, the resistance was twice as great, and the velocity twice as slow, as when the 12 ounce weight was employed. But by comparing the results in column 2d with the corresponding results in column 3d, it appears, that when the 24 ounce weight was employed, and the velocity was only one-half of what it was when the 12 ounce weight was used, the effect was more than one-half, the numbers in the 3d column being more than one-half the numbers in the 2d. Hence we may conclude, that the slower an overhot-wheel moves, the greater will be its performance.

262. These experiments of Deparcieux presented such unexpected results, as to induce other philosophers to examine them with care. The chevalier d'Arcy, in particular, considered them attentively. He maintained that there was a determinate velocity when the effect of the wheel reached its maximum; and he has shewn, maintaining by comparing the experiments of Deparcieux with his own formulae, that the overhot wheel which Deparcieux employed never moved with such a small velocity as corresponded with the maximum effect, and that effect. if he had increased the diameter of his cylinders, or the magnitude of the weights, his own experiments would have exhibited the degree of velocity, when the effect was the greatest possible.

263. The reasoning of the chevalier d'Arcy is completely confirmed by the experiments of Smeaton. This celebrated engineer concludes with Deparcieux that, ceteris paribus, the less the velocity of the wheel, the greater will be its effect. But he observes, on the contrary, that when the wheel of his model made about 30 turns in a minute, the effect was nearly the greatest; when it made 30 turns, the effect was diminished about one-twentieth part; and that when it made 40 it was diminished about one-fourth; when it made less than 18½ turns, its motion was irregular, and when it was loaded so that it could not make 18 turns, the wheel was overpowered by its load. Mr Smeaton likewise observes, that when the circumferences of overshot wheels, whether high or low, move with the velocity of three feet per second, and when the other parts of the work are properly adapted to it, they will produce the greatest possible effect. He allows, however, that high wheels may deviate farther from this rule before losing their power than low ones can be permitted to do; and assures us that he has seen a wheel 24 feet On Water-Wheels high moving at the rate of six feet per second, without loosing any considerable part of its power, and likewise a wheel 33 feet high moving very steadily and well with a velocity but little exceeding two feet.

264. The experiments of the abbé Boffut may also be brought forward in support of the same reasoning, by the experiments of Boffut. He employed a wheel 3 feet in diameter, furnished with 48 buckets, having each three inches of depth, and four inches of width. The canal which conveyed the water into the buckets was perfectly horizontal, and was five inches wide. It furnished uniformly 1194 cubic inches of water in a minute. The resistance to be overcome was a variety of weights fixed to the extremity of a cord, which, after passing over a pulley as in Deparcieux's experiments, wound round the cylindrical axle of the wheel. The diameter of this cylinder was two inches and seven lines, and that of the gudgeons or pivots of the wheel two lines and a half. The number of turns which the wheel made in a minute was not reckoned till its motion became uniform, which always happened when it had performed five or six revolutions. When the wheel was unloaded it made 40½ turns in a minute.

<table> <tr> <th>Number of pounds raised.</th> <th>Number of seconds in which the load was raised.</th> <th>Number of revolutions performed by the wheel.</th> <th>Effect of the wheel, or the product of the number of turns multiplied by the load.</th> </tr> <tr> <td>11</td> <td>60"</td> <td>11 2/3</td> <td>131 1/3</td> </tr> <tr> <td>12</td> <td>60</td> <td>11 5/6</td> <td>134 1/6</td> </tr> <tr> <td>13</td> <td>60</td> <td>10 2/3</td> <td>136 2/3</td> </tr> <tr> <td>14</td> <td>60</td> <td>9 7/8</td> <td>137 7/8</td> </tr> <tr> <td>15</td> <td>60</td> <td>9 1/8</td> <td>138 1/8</td> </tr> <tr> <td>16</td> <td>60</td> <td>8 3/8</td> <td>138 3/8</td> </tr> <tr> <td>17</td> <td>60</td> <td>8 9/16</td> <td>139 9/16</td> </tr> <tr> <td>18</td> <td>60</td> <td>7 1/8</td> <td>138</td> </tr> <tr> <td>19</td> <td colspan="3">The wheel turned but exceedingly slow.</td> </tr> <tr> <td>20</td> <td colspan="3">The wheel stopped though first put in motion by the hand to make it catch the water.</td> </tr> </table>

265. It appears evidently from the last column, which we have computed on purpose, that the effect increases as the velocity diminishes; but that the effect is a maximum when the number of turns is 8 9/16 in a minute, being then 139 9/16. When the velocity was farther diminished by adding an additional pound to the resistance, the effect was diminished to 138, and when the velocity was still less, the wheel ceased to move.

Now since the wheel was three feet in diameter, and 9.42 feet in circumference, the velocity of its circumference will be about one foot four inches per second, when it performs 8 9/16 turns in a minute, or when the maximum effect is produced. With Mr Smeaton's model, the maximum effect was produced when the velocity of the wheel's circumference was two feet per second. So that the experiments both of Smeaton and Boffut concur to prove, that the power of overshot wheels increases as the velocity diminishes; but that there is a certain velocity, between one and two feet per second, when the wheel produces a maximum effect. Since when the wheel was unloaded it turned 40½ times in a minute, and performed only 8 9/16 revolutions when its power was a maximum, the velocity of the wheel when unloaded will be to its velocity when the effect is the greatest, as five to one, nearly.

266. The chevalier de Borda maintains that an overshot wheel will raise through the height of the fall a quantity of water equal to that by which it is driven, and Albert Euler has shewn that the effect of these wheels is very much inferior to the momentum or force which impels them. It appears, however, from Mr Smeaton's experiments, 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 On Water-fall and the quantities of water expended were the least; but that it was as four to two when the heights 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 one. 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.

267. The theory of overshot wheels has been ably discussed by Albert Euler, and Lambert. The former of these philosophers has shown that the altitude of the wheel should be made as great as possible; that the buckets should be made as capacious as other circumstances will permit; that their form should be such as to convey the water as near the lowest point of the wheel as can be conveniently done; and that the motion of the wheel should be slow, that the buckets may be completely filled. He has likewise shewn that the effect of the wheel increases as its velocity is diminished; and that overshot wheels should be used only when there is a sufficient height of fall. The results of Lambert's investigations are less consonant with the experiments of Smeaton. By examining the following table, which contains these results, it will appear at once that he makes the diameter of the wheel much smaller than it ought to be.

<table> <tr> <th>Height of the fall, reckoning from the surface of the stream</th> <th>Radius of the wheel, reckoning from the extremity of the buckets.</th> <th>Width of the buckets.</th> <th>Depth of the buckets.</th> <th>Velocity of the wheel per second.</th> <th>Time in which the wheel performs one revolution.</th> <th>Turns of the mill-stone upon one of the wheel.</th> <th>Force of the water upon the buckets.</th> <th>The length of \( m_n \), in Fig. 6. Plate CCLXIX.</th> <th>The length of \( n_o \), in Fig. 6. Plate CCLXIX.</th> <th>Quantity of water required per second to turn the wheel.</th> </tr> <tr> <th>Feet.</th> <th>Feet.</th> <th>Feet.</th> <th>Feet.</th> <th>Feet.</th> <th>Seconds.</th> <th></th> <th>lbs. Avoir.</th> <th>Feet.</th> <th>Feet.</th> <th>Cub. Feet.</th> </tr> <tr> <td>7</td> <td>2.83</td> <td>1.00</td> <td>2.02</td> <td>5.27</td> <td>3.38</td> <td>8.45</td> <td>636</td> <td>0.33</td> <td>1.15</td> <td>10.55</td> </tr> <tr> <td>8</td> <td>3.22</td> <td>1.14</td> <td>1.44</td> <td>5.63</td> <td>3.61</td> <td>9.02</td> <td>595</td> <td>0.38</td> <td>1.32</td> <td>9.23</td> </tr> <tr> <td>9</td> <td>3.63</td> <td>1.27</td> <td>1.07</td> <td>5.94</td> <td>3.83</td> <td>9.57</td> <td>565</td> <td>0.42</td> <td>1.48</td> <td>8.21</td> </tr> <tr> <td>10</td> <td>4.04</td> <td>0.43</td> <td>0.82</td> <td>6.30</td> <td>4.04</td> <td>10.10</td> <td>531</td> <td>0.48</td> <td>1.65</td> <td>7.38</td> </tr> <tr> <td>11</td> <td>4.45</td> <td>0.57</td> <td>0.65</td> <td>6.60</td> <td>4.23</td> <td>10.57</td> <td>511</td> <td>0.52</td> <td>1.81</td> <td>6.71</td> </tr> <tr> <td>12</td> <td>4.86</td> <td>0.71</td> <td>0.52</td> <td>6.89</td> <td>4.42</td> <td>11.05</td> <td>486</td> <td>0.57</td> <td>1.98</td> <td>6.15</td> </tr> <tr> <td>1</td> <td>2</td> <td>3</td> <td>4</td> <td>5</td> <td>6</td> <td>7</td> <td>8</td> <td>9</td> <td>10</td> <td>11</td> </tr> </table>

Sect. II. On Breast Wheels.

268. A breast wheel partakes of the nature both of an overshot and an undershot wheel, and is driven partly by the impulse, but chiefly by the weight of the water. A water wheel of this kind is represented in fig. 6, where MC is the stream of water falling on the floatboard o, with a velocity corresponding to the altitude \( m_n \), and afterwards acting by its weight on the floatboards between o and B. The mill course o B is made concentric with the wheel, which is fitted to it in such a manner that very little water is allowed to escape at the sides and extremities of the floatboards. According to Mr Smeaton, the effect of a wheel driven in this manner is equal "to the effect of an undershot wheel 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 whose height is equal to the difference of level between the point where it strikes the wheel and the level of the tail water (M)." That is, the effect of the wheel A is equal to that of an undershot wheel driven by a fall of water equal to \( m_n \), added to that of an overshot wheel whose height is equal to \( n_D \).

269. Mr Lambert of the academy of sciences at Berlin (N) has shewn that when the floatboards arrive at the position \( o_p \), they ought to be horizontal: the point \( p \) should be lower than o, in order that the whole space between any two adjacent floatboards may be filled with water; and that Cm should be equal to the depth of the floatboards. He observes also, that a breast wheel should be used when the fall of water is above four feet in height, and below ten. The following table is calculated from Lambert's formulae, and exhibits at one view the results of his investigations.

(m) Smeaton on Mills, schol. p. 36. (n) Nouv. Mem. de l'Academie de Berlin, 1775, p. 71. TABLE for Breast Mills.

<table> <tr> <th>Height of the fall in feet = CD.</th> <th>Breadth of the float-boards.</th> <th>Depth of the float-boards.</th> <th>Radius of the water wheel reckoned from the extremity of the float-boards.</th> <th>Velocity of the wheel per second.</th> <th>Time in which the wheel performs one revolution.</th> <th>Turns of the mill-stone for one of the wheel.</th> <th>Force of the water upon the float-boards.</th> <th>The length of m, n, in Fig. 6. Plate CCLXIX.</th> <th>The length of m, n, in Fig. 6. Plate CCLXIX.</th> <th>Water required per second to turn the wheel.</th> </tr> <tr> <td></td> <td>Feet.</td> <td>Feet.</td> <td>Feet.</td> <td>Feet.</td> <td>Seconds.</td> <td></td> <td>lbs. Avoir.</td> <td>Feet.</td> <td>Feet.</td> <td>Gab. Feet.</td> </tr> <tr> <td>1</td> <td>0.17</td> <td>198.6</td> <td>0.75</td> <td>2.18</td> <td>1.92</td> <td>4.80</td> <td>1336</td> <td>0.08</td> <td>0.23</td> <td>74-30</td> </tr> <tr> <td>2</td> <td>0.34</td> <td>35.1</td> <td>1.50</td> <td>3.09</td> <td>2.72</td> <td>6.80</td> <td>1084</td> <td>0.15</td> <td>0.46</td> <td>37-15</td> </tr> <tr> <td>3</td> <td>0.51</td> <td>12.7</td> <td>2.26</td> <td>3.78</td> <td>3.33</td> <td>8.32</td> <td>886</td> <td>0.23</td> <td>0.68</td> <td>24-77</td> </tr> <tr> <td>4</td> <td>0.69</td> <td>6.2</td> <td>3.01</td> <td>4.36</td> <td>3.84</td> <td>9.60</td> <td>768</td> <td>0.30</td> <td>0.91</td> <td>18-57</td> </tr> <tr> <td>5</td> <td>0.86</td> <td>3.57</td> <td>3.76</td> <td>4.88</td> <td>4.28</td> <td>10.70</td> <td>686</td> <td>0.38</td> <td>1.14</td> <td>14-86</td> </tr> <tr> <td>6</td> <td>1.03</td> <td>2.25</td> <td>4.51</td> <td>5.35</td> <td>4.70</td> <td>11.70</td> <td>626</td> <td>0.46</td> <td>1.37</td> <td>12-38</td> </tr> <tr> <td>7</td> <td>1.20</td> <td>1.53</td> <td>5.26</td> <td>5.77</td> <td>5.08</td> <td>12.70</td> <td>581</td> <td>0.53</td> <td>1.60</td> <td>10-61</td> </tr> <tr> <td>8</td> <td>1.37</td> <td>1.10</td> <td>6.02</td> <td>6.17</td> <td>5.43</td> <td>13.58</td> <td>543</td> <td>0.60</td> <td>1.83</td> <td>9-29</td> </tr> <tr> <td>9</td> <td>1.54</td> <td>0.81</td> <td>6.77</td> <td>6.55</td> <td>5.76</td> <td>14.40</td> <td>512</td> <td>0.68</td> <td>2.05</td> <td>8-26</td> </tr> <tr> <td>10</td> <td>1.71</td> <td>0.77</td> <td>7.52</td> <td>6.99</td> <td>6.07</td> <td>15.18</td> <td>486</td> <td>0.76</td> <td>2.28</td> <td>7-43</td> </tr> </table>

270. It appears from the preceding table, that when the altitude of the fall of water is below three feet, there is such an unsuitable proportion between the depth and width of the floatboards, that a breast wheel cannot well be employed. It is also evident, on the other hand, that when the height of the fall approaches to ten feet, the depth of the floatboards is too small in relation to their width. These two extremes, therefore, ought to be avoided in practice. The eleventh column of the table contains the quantity of water necessary to drive the wheel; but the total quantity of water should always exceed this, by the quantity, at least, that escapes between the mill course and the sides and extremities of the floatboards (o).

271. The following are the dimensions of an excellent breast water wheel, differing very little from that which is represented in fig. 6. The water, however, instead of falling through the height c n which is 16 inches, is delivered on the floatboard o p, through an adjutage fix inches and a half high.—The height n D is four feet two inches; and therefore the whole height CD must be five feet and a half. The radius of the wheel AB is fix feet and a half, the breadth of each floatboard fix inches and a half, and their depth 28 inches. The point P of the wheel moves with the velocity of 7.88 feet in a second. The quantity of water discharged in a second is 3,266 cubic feet, and the force of impulsion upon the floatboards 3,56 pounds avoirdupois. On some occasions buckets have been used in breast wheels instead of floatboards; but this is evidently a disadvantage, as the height through which the water acts is diminished by the number of inches through which the water must fall in order to acquire the velocity of the wheel, and also by the vered fine of the arch above the lowest point of the wheel which may be considered as not loaded with water.

VOL. X. Part II.

(o) See Appendix to Ferguson's Lectures, vol. ii. p. 189. edit. 2d.

SECT. III. On Underfoot Wheels.

272. An underfoot wheel is a wheel with a number of floatboards dipoled on its circumference, which receive the impulse of the water conveyed to the lowest point of the wheel by an inclined canal. It is represented in fig. 1, where WW is the water wheel, and ABDFHKMV the canal or mill course, which conveys the water to K, where it strikes the plane floatboards n o, &c., and makes the wheel revolve about its axis.

273. 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 be sufficient to make AB 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 28, or as 25 to 12; and since the surface of the water Sb is bent from a b into a c before it is precipitated down the fall, it will be necessary to incurvate 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 distant from C, and raise the perpendiculars BE, DE. The point of intersection E will be the centre from which the arch BD is to be described; the radius being about 10 1/6 inches. Now, in order that the water may act more advantageously upon the floatboards 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 On Water-Wheels thus lose a great part of its velocity. It will be necessary, therefore, to make it move along FH, an arch of a circle to which DF and KH 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 four feet nine inches from the points F and H, and the centre of the arch FH will be determined. The distance HK, through which the water runs before it acts upon the wheel, should not be less than two or three 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 floatboards, KL should be about three inches, and the extremity o of the floatboard n o 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 arch 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 floatboards, and should be about nine inches distant from OP, a horizontal line passing through O the lowest point of the fall; for if OL were much less than nine inches, the water having spent the greatest part of its force in impelling the floatboard, 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 two yards. The canal which reconducts the water from the course of discharge to the river should slope about four inches in the first 200 yards, three 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 fuelled 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.

As it is of great importance that none of the water should escape either below the floatboards, or at their sides, without contributing to turn the wheel, the course of impulsion KV should be wider than the course at K, as represented in fig. 2, where CD the course of impulsion corresponds with LV in fig. 1. AB corresponds with HK and BC with KL. The breadth of the floatboards therefore should be wider than mn, and their extremities should reach a little below B, like no in fig. 1. When these precautions are properly taken, no water can escape without exerting its force upon the floatboards.

273. It has been disputed among philosophers, whether the wheel should be furnished with a small or a great number of floatboards. M. Pitot has shewn, that when the floatboards have different degrees of obliquity, the force of impulsion upon the different surfaces will be reciprocally as their breadths: Thus in under-fig. 3, the force of impulsion upon h e will be to the force upon DO, as DO to h e (p). Hence he concludes that the distance between the floatboards should be equal to one-half of the immersed arch, or that when one floatboard is at the bottom of the wheel, and perpendicular to the current, as DE, the preceding floatboard BC should be just leaving the stream, and the succeeding one FG just immersing into it. For when the three floatboards FG, DE, BC have the same position as in the figure, the whole force of the current NM will act upon DE when it is in the most advantageous position for receiving it, whereas, if another floatboard d e were inserted between FG and DE, the part ig would cover DO, and by thus substituting an oblique for a perpendicular surface, the effect would be diminished in the proportion of DO to ig. Hence it is evident that, upon this principle, the depth of the floatboard DE should be always equal to the versed sine of the arch EG (q).

274. Notwithstanding the plausibility of this reasoning, it will not be difficult to shew that it is destitute of foundation. It is evident from fig. 3, that when one rate of the floatboards DE is perpendicular to the stream, it Fig. 3, receives the whole impulse of the water in the most advantageous manner. But when it arrives at the position d e, and the succeeding one FG at the position fg, so that the angle eAg may be bisected by the perpendicular AE; the situation of these floatboards will be the most disadvantageous, for a great part of the water will escape between the extremities g and e of the floatboards without striking them, and the part ig of the floatboard, which is really impelled, is less than DE, and oblique to the current. The wheel, therefore, must move irregularly, sometimes quick and sometimes slow, according to the position of the floatboards with respect to the stream; and this inequality will increase with the arch plunged in the water. The reasoning of M. Pitot, indeed, is founded on the supposition, that if another floatboard fg were placed between FG and DF, it would annihilate the force of the water that impels it, and prevent any of the fluid from striking the corresponding part DO of the preceding floatboard. But this is not the case. For when the water has acted upon fg, it still retains a part of its motion, and after bending round the extremity g strikes DE with its remaining force. We are entitled, therefore, to conclude that advantage must be gained by using more floatboards than are recommended by Pitot.

275. It is evident from the preceding remarks, that in order to remove any inequality of motion in the wheel, and prevent the water from escaping below the extremities of the floatboards, the wheel should be furnished should be with the greatest possible number of floatboards, without loading it too much, or enfeebling the rim on which they are

(p) Mem. de l'Acad. Paris, 1729, 8vo. p. 359. (q) A table containing the number of floatboards for wheels of different diameters, and founded on this principle, has been computed by Mr Brewster. See Appendix to Ferguson's Lectures, vol. ii. p. 149. 2d Edit.

In Water-are fixed. This rule was first given by M. Dupetit Wheels. Vandin (r); and it is easily perceived, that if the millwright should err in using too many floatboards, this error in excess will be perfectly trifling, and that a much greater loss of power would be occasioned by an error in defect.

276. The section of the floatboards ought not to be rectangular like \( abne \) in fig. 3, but should be bevelled like \( abmc \). For if they were rectangular, the extremity \( bn \) would interrupt a portion of the water which would otherwise fall on the corresponding part of the preceding floatboard. In order to find the angle \( abm \), subtract from 180 degrees the number of degrees contained in the immersed arch CEG, and the half of the remainder will be the angle required.

277. It has been maintained by M. Pitot and other philosophers, that the floatboards should be a continuation of the radius, or perpendicular to the rim, as in fig. 1. This indeed is true in theory, but it appears from the most unquestionable experiments, that they should be inclined to the radius. This important fact was discovered by Depareieux in 1753, and proved by several experiments. When the floatboards are inclined, the water heaps up on their surface, and acts not only by its impulse but also by its weight. The same truth has also been confirmed by the abbé Boufut, the most accurate of whose experiments are contained in the following table. The wheel that was employed was immersed four inches vertically in the water, and it was furnished with 12 floatboards.

<table> <tr> <th>Inclination of the floatboard.</th> <th>Number of pounds raised.</th> <th>Time in which the load was raised in seconds.</th> <th>Number of turns made by the wheel.</th> </tr> <tr> <td>0</td> <td>40</td> <td>40</td> <td>13 1/8</td> </tr> <tr> <td>15</td> <td>40</td> <td>40</td> <td>14 1/8</td> </tr> <tr> <td>30</td> <td>40</td> <td>40</td> <td>14 1/8</td> </tr> <tr> <td>37</td> <td>40</td> <td>40</td> <td>14 1/8</td> </tr> <tr> <td>1</td> <td>2</td> <td>3</td> <td>4</td> </tr> </table>

278. It is obvious, from the preceding table, that the wheel made the greatest number of turns, or moved with the greatest velocity, when the number of floatboards was between 15 and 30. When the water-wheels are placed on canals that have little declivity, and in which the water can escape freely after its impulse upon the floatboards, it would be proper to make the floatboards a continuation of the radius. But when they move in an inclined mill-course, an augmentation of velocity may be expected from an inclination of the floatboards.

279. Having thus pointed out the most scientific method of constructing the wheel, and delivering the water upon its floatboards, we have now to determine the velocity with which it should move. It is evident, that the velocity of the wheel must be always less than that of the water which impels it, even when there is no On Water-work to be performed; for a part of the impelling power is necessarily spent in overcoming the inertia of the wheel and the resistance of friction. It is likewise obvious, that when the wheel has little or no velocity, its performance will be very trifling. There is, consequently, a certain proportion between the velocity of the water and the wheel, when its effect is a maximum. By the reasoning which is employed in the section on undershot wheels in the article WATER-WORKS, Parent and Pitot found, that a maximum effect was produced when the velocity of the wheel was one-third of the velocity of the water; and Desaguliers (s), Maclaurin (t), Lambert (u), and Atwood (x), have adopted their conclusions. In the calculus from which this result was deduced, it was taken for granted, that the momentum or force of water upon the wheel is in the duplicate ratio of the relative velocity, or as the square of the difference between the velocity of the water and that of the wheel. This supposition, indeed, is perfectly correct when the water impels a single floatboard; for as the number of particles which strike the float, impelling board in a given time, and also the momentum of these, are each as the relative velocity of the floatboards, the wheel wheel must be as the square of the relative velocity, that is, \( \frac{M}{R^2} \), \( M \) being the momentum, and \( R \) the relative velocity. But we have seen, in some of the preceding paragraphs, that the water acts on more than one floatboard at a time. Now the number of floatboards acted upon in a given time will be as the velocity of the wheel, or inversely as the relative velocity; for if you increase the relative velocity, the velocity of the water remaining the same, you must diminish the velocity of the wheel. Consequently, we shall have \( \frac{M}{R^2} \) or \( \frac{M}{R} \); that is, the momentum of the water acting upon the wheel, is directly as the relative velocity.

280. Let V be now the velocity of the stream, and F the force with which it would strike the floatboard at rest, and v the velocity of the wheel. Then the relative velocity will be \( V - v \); and since the velocity of the water will be to its momentum, or the force with which it would strike the floatboard at rest, as the relative velocity is to the real force which the water exerts against the moving floatboards, we shall have

\[ V : V - v = F : F \times \frac{V - v}{V} = F \times \frac{V - v}{V} \]

But the effect of the wheel is measured by the product of the momentum of the water and the velocity of the wheel, consequently the effect of the undershot wheel will be

\[ v \times \frac{F}{V} \times V - v = \frac{F}{V} \times V v - v^2 \]

Now this effect is to be a maximum, and therefore its fluxion must be equal to 0, that is, \( v \) being the variable quantity, \( V v - 2 v v = 0 \), or \( 2 v v = V v \). Dividing by \( v \), we have \( 2 v = \frac{V}{v} \).

(r) Memoires des Savans Etrangers, tom. i. (s) Desaguliers' Experimental Philosophy, vol. ii. p. 424. lext. 12. (t) Atwood on Rectilineal and Rotatory Motion, p. 275—284. (u) Maclaurin's Fluxions, art. 907. p. 728. (x) Nouve Memoires de l'Acad. Berlin, 1775, p. 63. On Water Wheels. \( V \), and \( v = \frac{V}{2} \), that is, the velocity of the wheel will be one-half the velocity of the fluid when the effect is a maximum.

281. This result, which was first obtained by the chevalier de Borda, has been amply confirmed by the experiments of Mr Smeaton. "The velocity of the stream (says he) varies at the maximum between one-third and one-half that of the water; but in all the cases in which most work is performed in proportion to the water expended, and which approach the nearest to the circumstances of great works, when properly executed, the maximum lies much nearer one-half than one-third, one half seeming to be the true maximum, if nothing were lost by the resistance of the air, the scattering of the water carried up by the wheel, &c."

282. A result, nearly similar to this, was deduced from the experiments of Boffut. He employed a wheel whose diameter was three feet. The number of floatboards was at one time 48, and at another 24, their width being five inches, and their depth fix. The experiments with the wheel, when it had 48 floatboards, were made in an inclined canal, supplied from a reservoir by an orifice two inches deep, the velocity being 300 feet in 27 seconds. The experiments with the wheel, when it had 24 floatboards, were made in a canal, contained between two vertical walls, 12 or 13 feet distant. The depth of the water was about seven or eight inches, and its mean velocity about 2740 inches in 40 seconds. The floatboards of the wheel were immersed about four inches in the stream.

<table> <tr> <th rowspan="2">Time in which the load is raised.</th> <th colspan="2">48 Floatboards</th> <th colspan="2">24 Floatboards</th> </tr> <tr> <th>No. of pounds raised.</th> <th>Number of turns made by the wheel.</th> <th>No. of pounds raised.</th> <th>Number of turns made by the wheel.</th> </tr> <tr><td>40</td><td>30 1/2</td><td>22 1/2</td><td>30</td><td>17 2/3</td></tr> <tr><td>40</td><td>31</td><td>22 1/2</td><td>35</td><td>16 2/3</td></tr> <tr><td>40</td><td>31 1/2</td><td>21 1/2</td><td>40</td><td>15 2/3</td></tr> <tr><td>40</td><td>32</td><td>21 1/2</td><td>45</td><td>14 2/3</td></tr> <tr><td>40</td><td>32 1/2</td><td>21 1/2</td><td>50</td><td>13 2/3</td></tr> <tr><td>40</td><td>33</td><td>21 1/2</td><td>55</td><td>12 2/3</td></tr> <tr><td>40</td><td>33 1/2</td><td>20 1/2</td><td>56</td><td>12 2/3</td></tr> <tr><td>40</td><td>34</td><td>20 1/2</td><td>57</td><td>12 2/3</td></tr> <tr><td>40</td><td>34 1/2</td><td>20 1/2</td><td>58</td><td>12 2/3</td></tr> <tr><td>40</td><td>35</td><td>19 1/2</td><td>59</td><td>12 2/3</td></tr> <tr><td>40</td><td>35 1/2</td><td>19 1/2</td><td>60</td><td>11 2/3</td></tr> <tr><td>40</td><td>36</td><td>18 1/2</td><td>61</td><td>11 2/3</td></tr> <tr><td>40</td><td>36 1/2</td><td>18 1/2</td><td>62</td><td>11 2/3</td></tr> <tr><td>40</td><td>37</td><td>18 1/2</td><td>63</td><td>11 2/3</td></tr> <tr><td>40</td><td>37 1/2</td><td>18 1/2</td><td>64</td><td>10 2/3</td></tr> <tr><td>40</td><td>38</td><td>18 1/2</td><td>65</td><td>10 2/3</td></tr> <tr><td>40</td><td>38 1/2</td><td>18 1/2</td><td>66</td><td>10 2/3</td></tr> </table>

283. As the effect of the machine is measured by the product of the load raised, and the time employed, it will appear, by multiplying the second and third columns, that the effect was a maximum when the load was 54 1/2 pounds, the wheel performing 20 1/2 revolutions in 40 seconds. By comparing the velocity of the centre of impulsion computed from the diameter of the wheel, and the number of turns which it makes in 40 seconds, with the velocity of the current, it will be found, that the velocity of the wheel, when its effect is the greatest possible, is nearly two-fifths that of the stream. From the two last columns of the table, where the effect is a maximum when the load is 60 pounds, the same conclusion may be deduced.

284. The proper velocity of the wheel being thus established, we shall proceed to point out the method of constructing a mill-wright's table for undershot wheels, taking it for granted, that the velocity of the wheel should be one-half the velocity of the stream, and that water moves with the same velocity as falling bodies.

1. Find the perpendicular height of the fall of water above the bottom of the mill-course, and having diminished this number by one-half the depth of the water at K, call that the height of the fall.

2. Since bodies acquire a velocity of 32.174 feet, by falling through the height of 16.087 feet; and as the velocities of falling bodies are as the square roots of the heights through which they fall, the square root of 16.087 will be to the square root of the height of the fall as 32.174 to a fourth number, which will be the velocity of the water. Therefore the velocity of the water may be always found by multiplying 32.174 by the square root of the height of the fall, and dividing that product by the square root of 16.087. Or it may be found more easily by multiplying the height of the fall by the constant quantity 64.348 = 2 × 32.174, and extracting the square root of the product. This root, abstracting from the effects of friction, will be the velocity of the water required.

3. Take one-half the velocity of the water, and it will be the velocity which must be given to the floatboards, or the number of feet they must move through in a second, in order to produce a maximum effect.

4. Divide the circumference of the wheel by the velocity of its floatboards per second, and the quotient will be the number of seconds in which the wheel revolves.

5. Divide 60 by the number last found, and the quotient will be the number of turns made by the wheel in a minute.—Or the number of revolutions performed by the wheel in a minute may be found, by multiplying the velocity of the floatboards by 60, and dividing the product by the circumference of the wheel.

6. Divide 92, the number of revolutions which a millstone, five feet diameter, should make in a minute, by the number of revolutions made by the wheel in a minute; and the quotient will be the number of turns which the millstone ought to make for one revolution of the wheel.

7. Then as the number of revolutions of the wheel in a minute, is to the number of revolutions of the millstone in a minute, so must the number of leaves in the trundle be to the number of teeth in the wheel, in the nearest whole numbers that can be found.

8. Multiply the number of revolutions performed by the wheel in a minute, by the number of revolutions made by the millstone for one of the wheel, and the product will be the number of revolutions made by the millstone in a minute.

285. By these rules, the following table has been computed.

On Water-Wheels.

computed for a water wheel fifteen feet in diameter, which is a good medium size, the millstone being seven feet in diameter, and revolving 90 times in a minute.

TABLE I. A New Mill-Wright's Table, in which the Velocity of the Wheel is one-half the Velocity of the Stream, the effects of Friction not being considered.

<table> <tr> <th rowspan="2">Height of the fall of water.</th> <th colspan="2">Velocity of the water per second, friction not being considered.</th> <th colspan="2">Velocity of the wheel per second, being one-half that of the water.</th> <th colspan="2">Revolutions of the wheel per minute, its diameter being 15 feet.</th> <th colspan="2">Revolutions of the mill-stone per minute, one of the wheel.</th> <th colspan="2">Teeth in the wheel and flaves in the trundle.</th> <th colspan="2">Revolutions of the mill-stone per minute, by these flaves and teeth.</th> </tr> <tr> <th>Feet.</th> <th>100 parts of a foot.</th> <th>Feet.</th> <th>100 parts of a foot.</th> <th>Revols.</th> <th>100 parts of a revol.</th> <th>Revols.</th> <th>100 parts of a revol.</th> <th>Tooths.</th> <th>Revols.</th> <th>100 parts of a revol.</th> </tr> <tr><td>1</td><td>8.02</td><td>4.01</td><td>5.10</td><td>17.65</td><td>106</td><td>6</td><td>92.01</td></tr> <tr><td>2</td><td>11.34</td><td>5.67</td><td>7.22</td><td>12.47</td><td>87</td><td>7</td><td>93.03</td></tr> <tr><td>3</td><td>13.89</td><td>6.95</td><td>8.85</td><td>10.17</td><td>81</td><td>8</td><td>90.00</td></tr> <tr><td>4</td><td>16.04</td><td>8.02</td><td>10.20</td><td>8.82</td><td>79</td><td>9</td><td>89.06</td></tr> <tr><td>5</td><td>17.91</td><td>8.97</td><td>11.43</td><td>7.97</td><td>71</td><td>9</td><td>89.05</td></tr> <tr><td>6</td><td>19.65</td><td>9.82</td><td>12.50</td><td>7.20</td><td>65</td><td>9</td><td>89.08</td></tr> <tr><td>7</td><td>21.22</td><td>10.61</td><td>13.51</td><td>6.66</td><td>60</td><td>9</td><td>89.08</td></tr> <tr><td>8</td><td>22.69</td><td>11.34</td><td>14.45</td><td>6.23</td><td>56</td><td>9</td><td>89.02</td></tr> <tr><td>9</td><td>24.06</td><td>12.03</td><td>15.31</td><td>5.88</td><td>53</td><td>9</td><td>89.02</td></tr> <tr><td>10</td><td>25.37</td><td>12.69</td><td>16.17</td><td>5.57</td><td>50</td><td>10</td><td>92.06</td></tr> <tr><td>11</td><td>26.60</td><td>13.30</td><td>16.95</td><td>5.31</td><td>53</td><td>10</td><td>92.00</td></tr> <tr><td>12</td><td>27.79</td><td>13.93</td><td>17.70</td><td>5.08</td><td>51</td><td>10</td><td>89.91</td></tr> <tr><td>13</td><td>28.92</td><td>14.46</td><td>18.41</td><td>4.89</td><td>49</td><td>10</td><td>90.02</td></tr> <tr><td>14</td><td>30.01</td><td>15.01</td><td>19.16</td><td>4.71</td><td>47</td><td>10</td><td>92.00</td></tr> <tr><td>15</td><td>31.07</td><td>15.53</td><td>19.80</td><td>4.55</td><td>48</td><td>11</td><td>92.09</td></tr> <tr><td>16</td><td>32.09</td><td>16.04</td><td>20.40</td><td>4.43</td><td>44</td><td>10</td><td>89.96</td></tr> <tr><td>17</td><td>33.07</td><td>16.54</td><td>21.05</td><td>4.28</td><td>47</td><td>11</td><td>97.29</td></tr> <tr><td>18</td><td>34.03</td><td>17.02</td><td>21.66</td><td>4.16</td><td>50</td><td>12</td><td>97.10</td></tr> <tr><td>19</td><td>34.97</td><td>17.48</td><td>22.26</td><td>4.04</td><td>44</td><td>11</td><td>89.93</td></tr> <tr><td>20</td><td>35.97</td><td>17.99</td><td>22.86</td><td>3.94</td><td>48</td><td>12</td><td>93.07</td></tr> </table>

1 2 3 4 5 6 7

"de Borda supposes it never to exceed three-eighths; On Water-Wheels. and Mr Smeaton and the abbé Bofluit found two-fifths to be the proper medium (r). With three-sevenths, therefore, as the best medium, which differs only \( \frac{1}{7} \)th from \( \frac{2}{5} \), the numbers in the following table have been computed. In Table I, the water was supposed to move with the same velocity as falling bodies, but owing to its friction on the mill-course, &c. this is not exactly the case. We have therefore deduced the velocity of the water in column second, from the following formula, \( V = \sqrt{\frac{172}{3}} \times R b - \frac{H h}{2} \), Fig. 1.

in which V is the velocity of the water, R b the absolute height of the fall, and H h the depth of the water at the bottom of the course. The formula is founded on the experiments of Bofluit, from which it appears, that if a canal be inclined one-tenth part of its length, this additional declivity will restore that velocity to the water which was destroyed by friction."

TABLE II. A New Mill-Wright's Table, in which the Velocity of the Wheel is three-sevenths of the Velocity of the Water, and the effects of Friction on the Velocity of the stream reduced to computation.

<table> <tr> <th rowspan="2">Height of the fall of water.</th> <th colspan="2">Velocity of the water per second, friction being considered.</th> <th colspan="2">Velocity of the wheel per second, being 3-5ths that of the water.</th> <th colspan="2">Revolutions of the wheel per minute, its diameter being 15 feet.</th> <th colspan="2">Revolutions of the mill-stone per minute, one of the wheel.</th> <th colspan="2">Teeth in the wheel and flaves in the trundle.</th> <th colspan="2">Revolutions of the mill-stone per minute, by these flaves and teeth.</th> </tr> <tr> <th>Feet.</th> <th>100 parts of a foot.</th> <th>Feet.</th> <th>100 parts of a foot.</th> <th>Revols.</th> <th>100 parts of a revol.</th> <th>Revols.</th> <th>100 parts of a revol.</th> <th>Tooths.</th> <th>Revols.</th> <th>100 parts of a revol.</th> </tr> <tr><td>1</td><td>7.62</td><td>3.27</td><td>4.16</td><td>21.63</td><td>130</td><td>6</td><td>89.98</td></tr> <tr><td>2</td><td>10.77</td><td>4.62</td><td>5.88</td><td>15.31</td><td>92</td><td>6</td><td>90.02</td></tr> <tr><td>3</td><td>13.20</td><td>5.66</td><td>7.20</td><td>22.50</td><td>100</td><td>8</td><td>90.00</td></tr> <tr><td>4</td><td>15.24</td><td>5.53</td><td>8.32</td><td>10.81</td><td>97</td><td>9</td><td>89.94</td></tr> <tr><td>5</td><td>17.24</td><td>7.30</td><td>9.28</td><td>9.70</td><td>97</td><td>11</td><td>90.02</td></tr> <tr><td>6</td><td>18.07</td><td>8.00</td><td>10.19</td><td>8.83</td><td>97</td><td>12</td><td>89.98</td></tr> <tr><td>7</td><td>20.15</td><td>8.64</td><td>10.99</td><td>8.19</td><td>90</td><td>11</td><td>90.01</td></tr> <tr><td>8</td><td>21.56</td><td>9.24</td><td>11.76</td><td>7.65</td><td>84</td><td>11</td><td>89.96</td></tr> <tr><td>9</td><td>22.86</td><td>9.80</td><td>12.47</td><td>7.22</td><td>72</td><td>10</td><td>90.03</td></tr> <tr><td>10</td><td>24.10</td><td>10.33</td><td>13.15</td><td>6.84</td><td>82</td><td>12</td><td>89.95</td></tr> <tr><td>11</td><td>25.27</td><td>10.83</td><td>13.79</td><td>6.53</td><td>85</td><td>13</td><td>90.05</td></tr> <tr><td>12</td><td>26.42</td><td>11.31</td><td>14.40</td><td>6.25</td><td>72</td><td>12</td><td>90.00</td></tr> <tr><td>13</td><td>27.47</td><td>11.77</td><td>14.99</td><td>6.00</td><td>72</td><td>12</td><td>89.94</td></tr> <tr><td>14</td><td>28.51</td><td>12.22</td><td>15.56</td><td>5.78</td><td>75</td><td>13</td><td>90.04</td></tr> <tr><td>15</td><td>29.52</td><td>12.65</td><td>16.13</td><td>5.58</td><td>87</td><td>12</td><td>90.01</td></tr> <tr><td>16</td><td>30.48</td><td>13.06</td><td>16.63</td><td>5.41</td><td>65</td><td>12</td><td>89.97</td></tr> <tr><td>17</td><td>31.42</td><td>13.46</td><td>17.14</td><td>5.25</td><td>63</td><td>12</td><td>89.99</td></tr> <tr><td>18</td><td>32.33</td><td>13.86</td><td>16.65</td><td>5.13</td><td>61</td><td>12</td><td>90.01</td></tr> <tr><td>19</td><td>33.22</td><td>14.24</td><td>18.13</td><td>4.96</td><td>61</td><td>13</td><td>89.92</td></tr> <tr><td>20</td><td>34.17</td><td>14.64</td><td>18.64</td><td>4.83</td><td>58</td><td>12</td><td>89.84</td></tr> </table>

1 2 3 4 5 6 7

(r) The great hydraulic machine at Marly was found to produce a maximum effect, when its velocity was two-fifths that of the stream. 287. In order that the wheel may move with a velocity duly adjusted to that of the current, we would not advise the mechanic to trust to the second column of Table II. for the true velocity of the stream, or to any theoretical results, even when deduced from formulae founded on experiments. Boffit, with great justice, remarks, that "it would not be exact in practice to compute the velocity of a current from its declivity. This velocity ought to be determined by immediate experiment in every particular case." Let the velocity of the water, therefore, where it strikes the wheel, be determined by the method in the following paragraph. With this velocity, as an argument, enter column second of either of these tables, according as the velocity of the wheel is to be one-half or three-sevenths that of the stream, and take out the other numbers from the table.

288. Various methods have been proposed by different philosophers for measuring the velocity of running water; the method, by floating bodies, which Mariotte (z) employed, the bent tube of Pitot (a), the regulator of Guglielmini (b), the quadrant (c), the little wheel (d), and the method proposed by the Abbé Mann (e), have each their advantages and disadvantages. The little wheel was employed in the experiments of Boffit. It is the most convenient mode of determining the superficial velocity of the water; and, when constructed in the following manner, will be more accurate, it is hoped, than any instrument that has hitherto been used. 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 floatboards. This wheel moves upon a delicate screw a B, passing through its axle B b; 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 m a, by means of the index O h fixed at O, and moveable with the wheel, each division of the scale being equal to the breadth of a thread of the screw, and the extremity h of the index O h coinciding with the beginning 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 m n 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 it rest on the handles C, D; and screw the shoulder b of the wheel close to a, so that the indices may both point to o 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 b B) by the number of revolutions, and the product will be the number of feet through which the water moves in the given time. On account of the friction of the screw, the resistance of the air, and the weight of the wheel, its centre of impression will revolve with a little less velocity than that of the stream; but the diminution of velocity, arising from these causes, may be estimated with sufficient precision for all the purposes of the practical mechanic. (Appendix to Ferguson's Lectures, vol. ii. p. 177.)

289. It appears, from a comparison of the numerous Results of and accurate experiments of Mr Smeaton, that, in un-Smeaton's dershot wheels, the power employed to turn the wheel is to the effect produced as 3 to 1; and that the load which the wheel will carry at its maximum, is to the load which will totally stop it, as 3 to 4. The same experiments inform us, that the impulse of the water on the wheel, in the case of a maximum, is more than double of what is assigned by theory, that is, instead of four-sevenths of the column, it is nearly equal to the whole column. In order to account for this, Mr Smeaton observes, that the wheel was not, in this case, placed in an open river, where the natural current, after it had communicated its impulse to the float, has room on all sides to escape, as the theory supposes; but in a conduit or race, to which the float being adapted, the water could not otherwise escape than by moving along with the wheel. He likewise remarks, that when a wheel works in this manner, the water, as soon as it meets the float, receives a sudden check, and rises up against it like a wave against a fixed object; insomuch, that when the sheet of water is not a quarter of an inch thick before it meets the float, yet this sheet will act upon the whole surface of a float, whose height is three inches. Were the float, therefore, no higher than the thickness of the sheet of water, as the theory supposes, a great part of the force would be lost by the water dashing over it. In order to try what would be the effect of diminishing the number of floatboards, Mr Smeaton reduced the floatboards, which were originally 24 to 12. This change produced a diminution of the effect, as a greater quantity of water escaped between the floats and the floor. But when a circular sweep was adapted to the floor, and made of such a length that one float entered the curve before the preceding one quitted it, the effect came so near to the former, as to afford no hopes of increasing it by augmenting the number of floats beyond 24 in this particular wheel. Mr Smeaton likewise deduced, from his experiments, the following maxims.

1. That the virtual or effective head being the same, the effect will be nearly as the quantity of water expended.

2. That

(z) Traité du Mouvement des Eaux. (a) Mem. de l' Acad. Paris, 1732. (b) Aquarum Fluentium Mensura, lib. iv. (c) Boffit Traité d'Hydrodynamique, art. 654. (d) Id. id. art. 655. (e) Philosophical Transactions, vol. lxix. 2. That the expence of water being the same, the effect will be nearly as the height of the virtual or effective head.

3. That the quantity of water expended being the same, the effect is nearly as the square of the velocity.

4. The aperture being the same, the effect will be nearly as the cube of the velocity of the water.

290. We have hitherto supposed the floatboards, though inclined to the radius, to be perpendicular to the plane of the wheel. Undershot-wheels, however, have sometimes been constructed with floatboards inclined to the plane of the wheel. A wheel of this kind is represented in fig. 5, where AB is the wheel, and CDEFGH the oblique floatboards. The horizontal current MN is delivered on the floatboards, so as to strike them perpendicularly. On account of the size of the floatboards, 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 imprefion should be to the velocity of the water, as radius is to triple the fine of the angle by which the floatboards are inclined to the plane of the wheel. If this inclination, therefore, be 60°, the velocity of the wheel at the centre of imprefion 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. \( 60^\circ = \frac{\sqrt{3}}{2} \). When the inclination is 30°, the ratio of the velocities will be found to be as 2 to 3.

291. In wheels of this kind, the floats may also be advantageously inclined to the radius. In this case, the stream, which still strikes them perpendicularly, is inclined to the horizon. If the angle formed by the common fection of the wheel and floatboards with the radius of the wheel, be \( = m \); and if the angle by which the floatboards are inclined to the plane of the wheel be \( = n \), then the angle which the floatboards shouid form with the direction in which the wheel moves, will be \( = \text{Cof.} m \times \text{Sin.} n \). In order, therefore, that the stream may strike the floatboards with a perpendicular impulse, its inclination to the horizon must be \( = m \), and its inclination to the plane of the wheel \( = 90^\circ - n \). The lefs that the velocity of the water is, the greater should be the angle \( m \); for there is, in this cafe, no danger that the celerity of the wheel be too great. The area of the floatboards ought to be much greater than the fection of the current; and the interval between two adjacent floatboards shouid be so great, that before the one completely withdraws itself from the action of the water, the other should begin to receive its impulse.

292. 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. 6. AB is the large water-wheel which moves horizontally upon its arbor CD. This arbor passes through the immoveable millstone 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, with this difference only, that the part On Water-m B n C, fig. 2. of which KL in fig. 1. is a fection, instead of being rectilineal like mn, must be circular like mP, and concentric with the rim of the wheel, sufficient room being left between it and the tips of the floatboards for the play of the wheel. In this construction, where the water moves in a horizontal direction before it strikes the wheel, the floatboards should be inclined about 25° to the plane of the wheel, and the fame number of degrees to the radius, fo that the loweft and outermost fides of the floatboards may be farthest up the stream.

293. Instead of making the canal horizontal before it delivers the water on the floatboards, they are frequently inclined in such a manner as to receive the impule perpendicularly, and in the direction of the declivity of the mill-courfe. When this construction is adopted, the maximum effect will be produced when the velocity of the floatboards is not lefs than \( \frac{5.67\sqrt{H}}{2 \text{ Sin.} A'} \) where H represents the height of the fall, and A' the angle which the direction of the fall makes with a vertical line. But as the quantity \( \frac{5.67\sqrt{H}}{2 \text{ Sin.} A'} \) evidently increases as the fine of A decreases, it follows, that without lefening the effect of these wheels, we may diminish the angle A, and thus augment considerably the velocity of the floatboards, according to the nature of the machinery employed; whereas, in vertical wheels, there is only one determinate velocity which produces a maximum effect.

294. In the southern provinces of France, where horizontal wheels are generally employed, the floatboards are made of a curvilinear form, fo as to be concave towards the stream. The Chevalier de Borda obferves, that in theory a double effect is produced when the floatboards are concave; but that the effect is diminished in practice, from the difficulty of making the fluid enter and leave the curve in a proper direction. Notwithstanding this difficulty, however, and other defects which might be pointed out, horizontal wheels with concave floatboards are always superior to those in which the floatboards are plain, and even to vertical wheels, when there is a sufficient fall of water. When the floatboards are plane, the wheel is driven merely by the impule of the stream; but when they are concave, a part of the water acts by its weight and increafes the velocity of the wheel. If the fall of water be 5 or 6 feet, a horizontal wheel with concave floatboards may be erected, whose maximum effect will be to that of the ordinary vertical wheels as 3 to 2.

295. An advantage attending horizontal wheels is, that the water may be divided into several canals, and delivered upon several floatboards at the fame time. Each stream will heap up on its corresponding floatboard, and produce a greater effect than if the force of the water had been concentrated on a fingle floatboard. Horizontal wheels may be employed with greatest advantage when a small quantity of water falls through a considerable height.

296. It has been disputed among mechanical philosophers, whether overshot or underhot wheels produce the greatest effect. M. Belidor maintained that the former were inferior to the latter, while a contrary opinion was was entertained by Desaguliers. It appears, however, from Mr Smeaton's experiments, that in overhot wheels the power is to the effect nearly as 3 to 2 or as 5 to 4 in general, whereas in underhot wheels it is only as 3 to 1. The effect of overhot wheels therefore is nearly double that of underhot wheels, other circumstances being the same. In comparing the relative effects of water-wheels, the Chevalier de Borda remarks that overhot wheels will raise through the height of the fall, a quantity of water equal to that by which they are driven; that underhot vertical wheels will produce only three-eighths of this effect; that horizontal wheels will produce a little less than one-half of it when the floatboards are plain, and a little more than one-half of it when the floatboards have a curvilinear form.

Besant's Underhot Wheel.

297. The water-wheel invented by Mr Besant of Brompton is constructed in the form of a hollow drum, so as to resist the admission of the water. The floatboards are fixed obliquely in pairs on the periphery of the wheel, so that each pair may form an acute angle open at its vertex, while one of the floatboards extends beyond the vertex of the angle. A section of the water-wheel is represented in fig. 1, where AB is the wheel, CD its axis, and m n, o p the position of the floatboards. The motion of common underhot wheels is greatly retarded by the resistance which the tail-water and the atmosphere oppose to the ascending floatboards; but in Besant's wheel this resistance is greatly diminished, as the floats emerge from the stream in an oblique direction. Although this wheel is much heavier than those of the common construction, yet it revolves more easily upon its axis, as the stream has a tendency to make it float.

Conical Horizontal Wheel with Spiral Floatboards.

298. In Guyenne and Languedoc, in the south of France, a kind of conical horizontal wheel is sometimes employed for turning machinery. It is constructed in the form of an inverted cone AB, with spiral floatboards winding round its surface. The wheel moves on a vertical axis AB, in the building DD, and is driven chiefly by the impulse of the water conveyed by the canal C to the oblique floatboards, the direction of the current being perpendicular to the floatboards at the place of impact. When the impulsive force of the water is annihilated, it descends along the spirals, and continues to act by its weight till it reaches the bottom, when it is carried off by the canal M.

CHAP. II. On Machines driven by the Reaction of Water.

299. 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 though this principle has not yet been adopted in practice, it appears from theory, and from some detached experiments on a small scale, 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.

SECT. I. On Dr Barker's Mill.

300. This machine, which is sometimes called Pa-Dr Barker's mill, is represented in fig. 3, where A is the canal that conveys the water into the upright tube B, which communicates with the horizontal arm C. 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 d and e 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 underflood, 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 C will remain at rest. But as soon as you open the orifice, 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 prelure upon the circular inch opposite to the orifice still continues, the equilibrium will be destroyed, and the arm C will move in a retrograde direction.

301. The upright spindle D, on which the arm revolves, is fixed in the bottom of the arm, and screwed to it below by the nut g. It is fixed to the upright tube by two cros bars at f, 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 H. The lower quiescent millstone I rests upon the floor K, in which is the hole L, to let the meal pass into a trough about M. The bridge-tree GF, which supports the millstone, tube, &c, is moveable on a pin at h, 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 o. 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 to X, and a small wheel W fixed to its extremity, which will communicate its motion to any species of mechanism. An improvement on this machine by M. Mathon de la Cour, and some excellent observations on the subject by Professor Robison, will be found in the article WATER-WORKS.

302. Mr Waring of the American Philosophical Society, has given a theory of Barker's mill with the improvement of M. Mathon de la Cour, which he has strangely ascribed to a Mr Rumsey about 20 years after it was published in Ressier's Journal de Phyique, Jan. and August 1775. Contrary to every other philosopher, he makes the effect of the machine equal only to that of a good underhot-wheel, moved with the same quantity of water, falling though the same height. The fol-

Machines lowing rules, however, deduced from his calculus may be of use to those who may wish to make experiments on the effect of this interesting machine.

1. Make the arm of the rotatory tube or arm C, from the centre of motion to the centre of the aperture, of any convenient length, not less than one-third (one-ninth according to Mr Gregory (r), who has corrected some of Waring's numbers) of the perpendicular height of the water's surface above their centres.

2. Multiply the length of the arm in feet by .614, and take the square root of the product for the proper time of a revolution in seconds, and adapt the other parts of the machinery to this velocity; or, if the time of a revolution be given, multiply the square of this time by 1.63 for the proportional length of the arm.

3. Multiply together the breadth, depth, and velocity per second, of the race, and divide the last product by 18.47 (times 14.27 according to Mr Gregory) the square root of the height, for the area of either aperture.

4. Multiply the area of either aperture by the height of the fall of water, and the product by 41\( \frac{1}{2} \) pounds (55.775 according to Mr Gregory), for the moving force estimated at the centres of the apertures in pounds avoirdupois.

5. The power and velocity at the aperture may be easily reduced to any part of the machinery by the simplest mechanical rules.

303. Long after the preceding machine had been described in several of our English treatises on machines, Professor Segner published in his hydraulics, as an invention of his own, the account of a machine, differing from this only in form. MN was the axis of the machine, corresponding with DX in Barker's mill, and a number of tubes AB were also so arranged round this axis that their higher extremities A formed a circular superficies into which the water flowed from a reservoir. When the machine has this form, it has been shewn by Albert Euler that the maximum effect is produced when the velocity is infinite, and that the effect is equal to the power. As a considerable portion of the power, however, must be consumed in communicating to the fluid the circular motion of the tubes; and as the portion thus lost must increase with the velocity of the tube, the effect will in reality sustain a diminution from an increase of velocity.

Sect. II. Description of Albert Euler's Machine driven by the Reaction of the Water.

304. This machine consists of two vessels, the lowest of which EEFF is moveable round the vertical axis OO, while the higher vessel remains immoveable. The form of the lowest vessel, which is represented by itself in fig. 6. is similar to that of a truncated bell, which is fastened by the cross beams m, n to the axis O so as to move along with it. The annular cavity h h h h terminates at e e in several tubes e f, e f, e f, diverging from the axis. Through the lower extremities of these tubes, which are bent into a right angle, the water flowing from the cavity h h h h issues with a velocity due to the altitude of its surface in h, h, and produces by its reaction a rotatory and retrograde motion round the axis OO. The cavity of the ring h, h, receives the water from the superior vessel GGHH, similar to the inferior vessel in fig. 6. but not connected with the axis OO. This vessel has also an annular cavity PP, into which the water is conveyed from a reservoir by the canal R. Around the lower part HH of the cavity, this vessel is divided into several apertures I i, placed obliquely that the water may descend with proper obliquity into the inferior vessel. The width of the higher vessel at HH ought to be equal to the width of the lower vessel at EE, that the water which issues from the former may exactly fill the annular cavity h, h, h, h.

When the machine is constructed in this way, its maximum effect will be equal to the power, provided all its parts be proportioned and adjusted according to the results in the following table, computed from the formulae of Albert Euler. In the table,

Q = the quantity of water, or number of cubic feet of water furnished in a second. T = the time, or number of seconds in which the lower vessel revolves. B = the breadth of the annular orifice in inches.

TABLE for Mills driven by the Reaction of Water.

<table> <tr> <th>Height of the fall of water.</th> <th>Sum of the areas of all the orifices at f, f, f, &c.</th> <th>Sum of the areas of all the orifices at f, f, f, &c.</th> <th>Mean radius of the annular orifice HH.</th> <th>Difference between the altitude of the two vessels.</th> <th>Tangent of the inclination of the tubes to the horizon</th> </tr> <tr> <th>Feet.</th> <th>Square Feet.</th> <th>Square Inches.</th> <th>Feet.</th> <th>Inches.</th> <th></th> </tr> <tr><td>1</td><td>0.17888×Q</td><td>2.5739×Q</td><td>0.8897×T</td><td>1.7695^QQTTBB</td><td>0.38400^QTB</td></tr> <tr><td>2</td><td>0.12649×Q</td><td>18.214×Q</td><td>1.2582×T</td><td>0.8847^QQTTBB</td><td>0.19200^QTB</td></tr> <tr><td>3</td><td>0.103228×Q</td><td>14.872×Q</td><td>1.5410×T</td><td>0.5898^QQTTBB</td><td>0.12800^QTB</td></tr> <tr><td>4</td><td>0.08944×Q</td><td>12.880×Q</td><td>1.7794×T</td><td>0.4424^QQTTBB</td><td>0.09600^QTB</td></tr> <tr><td>5</td><td>0.08000×Q</td><td>11.520×Q</td><td>1.9894×T</td><td>0.3539^QQTTBB</td><td>0.07680^QTB</td></tr> <tr><td>6</td><td>0.07303×Q</td><td>10.516×Q</td><td>2.1793×T</td><td>0.2949^QQTTBB</td><td>0.06400^QTB</td></tr> <tr><td>7</td><td>0.06761×Q</td><td>9.736×Q</td><td>2.3540×T</td><td>0.2528^QQTTBB</td><td>0.05486^QTB</td></tr> <tr><td>8</td><td>0.06325×Q</td><td>9.107×Q</td><td>2.5165×T</td><td>0.2212^QQTTBB</td><td>0.04800^QTB</td></tr> <tr><td>9</td><td>0.05963×Q</td><td>8.586×Q</td><td>2.6691×T</td><td>0.1966^QQTTBB</td><td>0.04267^QTB</td></tr> <tr><td>10</td><td>0.05657×Q</td><td>8.146×Q</td><td>2.8135×T</td><td>0.1769^QQTTBB</td><td>0.03840^QTB</td></tr> <tr><td>11</td><td>0.05394×Q</td><td>7.767×Q</td><td>2.9508×T</td><td>0.1609^QQTTBB</td><td>0.03491^QTB</td></tr> <tr><td>12</td><td>0.05104×Q</td><td>7.436×Q</td><td>3.0820×T</td><td>0.1475^QQTTBB</td><td>0.03200^QTB</td></tr> <tr><td>13</td><td>0.04961×Q</td><td>7.144×Q</td><td>3.2078×T</td><td>0.1361^QQTTBB</td><td>0.02954^QTB</td></tr> <tr><td>14</td><td>0.04781×Q</td><td>6.885×Q</td><td>3.3290×T</td><td>0.1264^QQTTBB</td><td>0.02743^QTB</td></tr> <tr><td>15</td><td>0.04619×Q</td><td>6.651×Q</td><td>3.4458×T</td><td>0.1179^QQTTBB</td><td>0.02560^QTB</td></tr> <tr><td>16</td><td>0.04472×Q</td><td>6.440×Q</td><td>3.5588×T</td><td>0.1106^QQTTBB</td><td>0.02400^QTB</td></tr> <tr><td>17</td><td>0.04339×Q</td><td>6.248×Q</td><td>3.6683×T</td><td>0.1041^QQTTBB</td><td>0.02259^QTB</td></tr> <tr><td>18</td><td>0.04216×Q</td><td>6.072×Q</td><td>3.7747×T</td><td>0.0983^QQTTBB</td><td>0.02133^QTB</td></tr> </table>

The determinations in the preceding table are exhibited in a general manner, that the machine may be accommodated to local circumstances. The time of a revolution T, for instance, is left undetermined, because upon this time depends the magnitude of the machine; and T may be assumed of such a value that the dimensions of the machine may be suitable to the given place, or to the nature of the work to be performed.

395. In order to shew the application of the preceding table, let it be required to construct the machine when the height of the fall is five feet, and when the reservoir furnishes one cubic foot of water in a second. In this case Q = 1, and therefore, by column 3d, the sum of the areas of the orifices will be 11.52 square inches. Consequently, if there are twelve orifices, the area of each orifice will be \( \frac{11.52}{12} = 0.96 \) of a square inch. Suppose the time of a revolution to be = 1 second or T = 1, then the 4th column will give the mean radius of the annular orifice = 1.9894 feet, or nearly two feet. Let the breadth of the annular orifice or B = \( \frac{1}{2} \) an inch, then the difference between the altitude

of each vessel will be \( 0.3539 \times \frac{QQ}{TTBB} = 0.3539 \times \frac{1 \times 1}{\frac{1}{2} \times \frac{1}{2} \times 1 \times 1} = 0.3539 \times 4 = 1.4156 \) inches.

Now as the sum of the heights of the vessels must be always equal to the height of the fall, half that sum will in the present case be two feet fix inches; and since half the difference of their altitudes is 7-tenths of an inch, the altitude of the superior vessel will be two feet fix inches and seven-tenths, and that of the inferior vessel two feet five inches and three-tenths. It appears from the last column of the table, that the tangent of the inclination of the tubes is 0.1336, which corresponds with an angle of 8° 44'.

356. The theory of this machine has also been discussed by Leonard Euler in the Mem. de l' Acad. Berlin, vol. vi. p. 311.; and its application to all kinds of work has been pointed out in a subsequent paper, entitled, Application de la Machine Hydraulique de M. Segner à toutes sortes d'ouvrage, et de ses avantages sur les autres Machines Hydrauliques dont on le fera ordinairement, Mem. Acad. Berlin, tom. vii. 1752, p. 271. The results of Euler's analysis are not sufficiently practical for the use of the general reader. But it appears from his investigations, as well as from those of John Bernoulli and other philosophers, that the reaction of water is the most powerful way in which the force of that fluid can be employed.

357. It has often occurred to the writer of this article, that a very powerful hydraulic machine might be constructed by combining the impulse with the reaction of water. If the spout a, for example, instead of delivering the water into the higher vessel, were to throw it upon a number of curvilinear floatboards fixed on its circumference, and so formed as to convey the water easily into the spiral canals, we should have a machine something like the conical horizontal wheel in fig. 2, with spiral channels instead of spiral floatboards; and which would in some measure be moved both by the impulse, weight, and reaction of the water.

CHAP. III. On Machines for raising Water.

SECT. I. On Pumps.

358. The subject of pumps has been fully and ably discussed by Dr Robison under the article Pump, to which we must refer the reader for a complete view of the theory of the machine. In that article, however, a reference is made to the present for a description of the ancient pump of Ctesibius, and of those in common use to which it has given rise. To these subjects, therefore, we must now confine our attention.

359. The pump was invented by Ctesibius, a mathematician of Alexandria, who flourished under Ptolemy Ptochon, about 120 years before Christ. In its original state it is represented in fig. 1, where ABCD is a brass cylinder with a valve L in its bottom. It is furnished with a piston MK made of green wood, so as not to swell in water, and adjusted to the bore of the cylinder by the interposition of a ring of leather. The tube CI connects the cylinder ABCD with another tube NH, the bottom of which is furnished with a valve I opening upwards. Now when the extremity DC of the cylinder is immersed in water, and the piston MK elevated, the pressure of the water upon the valve I, from below will be proportioned to the depth below the surface (q1). The valve will therefore open and admit the water into the cylinder. But when the piston is depressed, it will force the water into the tube CH, and through the valve I into the tube NH. As soon as the portion of water that was admitted into the cylinder ABCD, is thus impelled into the tube NH, the valve I will close. A second elevation of the piston will admit another quantity of fluid into the cylinder, and a second depression will force it into the tube NH; so that, by continuing the motion of the piston, the water may be elevated to any altitude in the tube. From this pump of Ctesibius are derived the three kinds of pumps now commonly used, the sucking, the forcing, and the lifting pump.

360. The common sucking pump is represented in Description fig. 2, where ICBL is the body of the pump immersed in the water at A. The moveable piston DG ing pump, is composed of the piston rod D d, the piston or bucket Fig. 2. G, and the valve a: The bucket H which is fixed to the body of the pump, is likewise furnished with a valve b, which, like the valve a, should by its own weight lie close upon the hole in the bucket till the working of the engine commences. The valves are made of brads, and have their lower surface covered with leather, in order to fit the holes in the bucket more exactly. The moveable bucket G is covered with leather, so as to suit exactly the bore of the cylinder, and to prevent any air from escaping between it and the pump. The piston DG may be elevated or depressed by the lever DQ, whose fulcrum is r, the extremity of the bent arm R r.

361. Let us now suppose the piston G to be depressed so that its inferior surface may rest upon the valve b, operation. Then if the piston G be raised to C, there would have been a vacuum between H and G if the valve b were immoveable. But as the valve b is moveable, and as the pressure of the air is removed from its superior surface, the air in the tube HL will, by its elasticity, force open the valve b, and expand itself through the whole cavity LC. This air, however, will be much rarer than that of the atmosphere; and since the equilibrium between the external air and that in the tube LH is destroyed by the rarefaction of the latter, the pressure of the atmosphere on the surface of the water in the vessel K will predominate, and raise the water to about e in the suction pipe HL, so that the air formerly included in the space LC will be condensed to the same state as that of the atmosphere. The elasticity of the air both above and below the valve b being now equal, that valve will fall by its own weight.—Let the piston DG be now depressed to b. The air would evidently resist its descent, did not the valve a open and give a free exit to the air in the space CH, for it cannot escape through the inferior valve b. When the piston reaches b, the valve a will fall by its weight; and when the piston is again elevated, the incumbent air will press the valve a firmly upon its orifice. During the second ascent of the piston to C the valve b will rise, the air between e H will rush into HC; and in consequence of its rarefaction, and inability to counteract the pressure of the atmosphere, the water will rise to f. In the same way it may be shewn, that at the next stroke of the piston the water will rise through the box H to B, and then the valve b which was raised by it will fall when the bucket G is at C. Upon depressing the bucket G again, the water cannot be driven through the valve b, which is pressed to its orifice by the water above it. At the next ascent of the piston a new quantity of water will rise through H, and follow the piston to C. When the piston again descends, the valve a will open; and as the water between C and H cannot be pushed through the valve b, it will rise through a, and have its surface at C when the piston G is at b; but when the piston rises, the valve a being shut by the water above it, this water will be raised up towards I, and issue at the pipe F. A new quantity of water will rush through H and fill the space HC; consequently, the surface of the fluid will always remain at C, and every succeeding elevation of the piston from b to C will make the column of water CH run out at the pipe F.

312. As the water rises in the pipe CL solely by the prelure of the atmosphere; and as a column of water, 33 feet high, is equal in weight to a column of air of the same base, reaching from the earth's surface to the top of the atmosphere, the water in the vessel K will not follow the piston G to a greater altitude than 33 feet; for when it reaches this height, the column of water completely balances, or is in equilibrium with, the atmosphere, and therefore cannot be raised higher by the prelure of the external air.

313. The forcing pump is represented in fig. 3, where Dd is the piston attached to a solid plunger g, adjusted to the bore of the pipe BC by the interposition of a ring of leather. The rectangular pipe MMN communicates with the tube BC by the cavity round H; and its upper extremity P is furnished with a valve a opening upwards. An air-vessel KK is fastened to R, and the tube FGI is introduced into it so as to reach as near as possible to the valve a.—Let us now suppose the plunger Dg to be depressed to b. As soon as it is elevated to C the air below it will be rarefied, and the water will ascend through the valve b in the same way as in the sucking pump, till the pipe is filled to C. The valve b will now be shut by the weight of the incumbent water; and therefore when the plunger Dg is depressed, it will force the water between C and b through the rectangular pipe MMN, into the air vessel KK. Before the water enters the air vessel, it opens the valve a, which shuts as soon as the plunger is again raised, because the prelure of the water upon its under side is removed. In this way the water is driven into the air vessel by repeated strokes of the plunger, till its surface is above the lower extremity of the pipe IG. Now, as the air in the vessel KK has no communication with the external air when the water is above I, it must be condensed more and more, as new quantities of water are injected. It will therefore endeavour to expand itself, and by pressing upon the surface H of the water in the air vessel, it will drive the water through the tube IG, and make it issue at F in a continued stream, even when the plunger is rising to C. If the pipe GHI were joined to the pipe MMN at P, without the intervention of an air vessel, the stream of water would issue at F only when the plunger was depressed.

314. The lifting pump, which is only a particular modification of the forcing pump, is represented in fig. 4. The barrel AB is fixed in the immovable frame KILM, the lower part of which is immersed in the water to be raised. The frame GEQHO consists of two strong iron rods EQ, GH which move through holes in IK and LM, the upper and lower ends of the pump. To the bottom GH of this frame is fixed an inverted piston with its bucket and valve uppermost at D. An inclined branch KH, either fixed to the top of the barrel, or moveable by a ball and socket, as represented at F, must be fitted to the barrel so exactly as to resist the admission both of air and water. The branch KH is furnished with a valve C opening upwards. Let the pump be now plunged in the water to Mode of the depth of D. Then if the piston frame be thrust down into the fluid, the piston will descend, and the water by its upward prelure will open the valve at D and gain admission above the piston. When the piston frame is elevated, it will raise the water above D along with it, and forcing it through the valve, it will be carried off by the spout.

315. An ingenious pump, invented by De la Hire, is represented in fig. 5. It raises water equally quick by pump descent as by the ascent of the piston. The pipes B, C, E, F, all communicate with the barrel MD, and Fig. 5 have each a valve at their top, viz. at b, S, e, f. The piston rod LM and plunger K never rise higher than K, nor descend lower than D, KD being the length of the stroke. When the plunger K is raised from D to K, the prelure of the atmosphere forces the water through the valve b, and fills the barrel up to the plunger, in the very same way as in the forcing pump. When the plunger K is depressed to D, it forces the water between K and b up the pipe E and through the valve e into the box G, where it issues at the orifice O. During the descent of the plunger K the valve f falls, and covers the top of the pipe F; and as the piston rod LM moves in a collar of leather at M, and is air-tight, the air above the plunger, between Q and M, will be rarefied, and likewise the air in the pipe CS, which communicates with the rarefied air by the valve S. The prelure of the air therefore will raise the water in CS, force it through the valve S, and fill the space above the plunger, expelling the rarefied air through the valve f. When the piston is raised from D to K, it will force the water through the bent pipe F into the box G, so that the same quantity of water will be discharged at O through the pipe F, during the ascent of the piston, as was discharged through the pipe E during the piston's descent. Above the pipe O is a close air-vessel P, so that when the water is driven above the spout O, it compresses the air in the vessel P, and this air acting by its elasticity on the surface of the water, forces it out at O in a constant and nearly equal stream. As the effect of the machine depends on a proper proportion between the height O of the spout above the surface of the well, and the diameter of the barrel, the following table will be of use to the practical mechanic. <table> <tr> <th>Height of the spout O above the well.</th> <th>Diameter of the barrel D.</th> <th>Height of the spout O above the well.</th> <th>Diameter of the barrel D.</th> </tr> <tr> <td>Feet.</td> <td>Inches.</td> <td>Feet.</td> <td>Inches.</td> </tr> <tr> <td>10</td> <td>6.9</td> <td>60</td> <td>2.8</td> </tr> <tr> <td>15</td> <td>5.6</td> <td>65</td> <td>2.7</td> </tr> <tr> <td>20</td> <td>4.9</td> <td>70</td> <td>2.6</td> </tr> <tr> <td>25</td> <td>4.4</td> <td>75</td> <td>2.5</td> </tr> <tr> <td>30</td> <td>4.0</td> <td>80</td> <td>2.5</td> </tr> <tr> <td>35</td> <td>3.7</td> <td>85</td> <td>2.4</td> </tr> <tr> <td>40</td> <td>3.5</td> <td>90</td> <td>2.3</td> </tr> <tr> <td>45</td> <td>3.3</td> <td>95</td> <td>2.2</td> </tr> <tr> <td>50</td> <td>3.1</td> <td>100</td> <td>2.1</td> </tr> </table>

When the proportions in the preceding table are observed, a man of common strength will raise water much higher than he could do with a pump of the common construction.

316. A very simple pump which furnishes a continued stream is represented in fig. 6. It was invented by a Mr Noble, and consists of a working barrel AB with two pistons C and B, which are moved up and down alternately by the rods fixed to the lever EMN. The rod of the piston B passes through the piston C, and the piston C moves upon the rod AB. When the piston rod B is depressed and elevated, it will make the water rise in the barrel A, in the same way as in the fucking pump, whether the valve C be moveable or not. Let us now suppose that the water is raised to A. Then if the piston B is elevated by depressing the extremity N of the lever, the water at A will be raised higher in the barrel, and issue at the spout P, and when the same piston B is depressed by elevating the end N of the lever, the piston C is evidently raised, and the water above it will be expelled at P. This pump, therefore, will give a continued stream, for as the pistons ascend and descend alternately, one of them must always be forcing the water out at P. The pistons are elevated and depressed by means of toothed arches, c and d, working in the teeth of a rack, at the extremities a, b of the piston rod.

317. The pump invented by Mr Buchanan is shewn in fig. 7. In the vertical section DGA, A is the suction barrel, D the working barrel, E the piston, G the spout, B the inner valve, and C the outer valve. These valves are of the kind called clack valves, and have their hinges generally of metal. It is easily seen that when the piston E is raised, the water will rise through the suction barrel A, into the working barrel D, in the same way as in the fucking pump; and that when the piston E is depressed, it will force the water between it and the valve B, through the valve C, and make it issue at G. The points of difference between this pump and those of the common form, are,—that it discharges the water below the piston, and has its valves lying near each other. Hence the sand or mud which may be in the water, is discharged without injuring the barrel or the piston leathers; and as the valves B, C may be of any size, they will transmit, without being choked, any rubbish which may rise in the suction barrels. If any obstruction should happen to the valves, they are within the reach of the workman's hand, and may be cleared without taking the pump to pieces. This simple machine may be quickly converted into a fire engine, by adding the air-vessel H, which is screwed like a hose-pipe, and by fixing in the spout G a perforated stopple fitted to receive such pipes as are employed in fire engines. When these additions are made, the water, as in the case of the forcing pump, will be driven into the air vessel H, and repelled through the perforated stopple G, by the elasticity of the included air.

318. A simple method of working two pumps at once Balance- by means of a balance, is exhibited in fig. 8. where AB pump. is the balance, having a large iron ball at each end, Figs. 8 & 9. placed in equilibrium on the two spindles C, see fig. 9. The person who works the pump stands on two boards I, I, nailed to two cross pieces fastened to the axis of the machine, and supports himself by a cross bar D joined to the two parts D, E. At the distance of ten inches on each side of the axis are suspended the iron rods M, N, to which the pistons are attached. The workman, by bearing alternately on the right and left foot, puts the balance in motion. The pistons M, N are alternately elevated and depressed, and the water raised in the barrel of each, is driven into the pipe HH, in which it is elevated to a height proportional to the diameter of the valves, and the power of the balance. In order to make the oscillations of the balance equal, and prevent it from acquiring too great a velocity, iron springs F, G are fixed to the upright posts, which limit the length of its oscillations.

319. The chain pump is represented in fig. 1. It consists of a chain MTHG, about 30 feet long, carrying pump. a number of flat pistons M, N, O, P, Q, which are made to revolve in the barrels ABCD and GH, by driving the wheel F. When the flat pistons are at the lower part of the barrel T, they are immersed in the water RR, and as they rise in the barrel GH, they bring up the water along with them into the reservoir MG, from which it is conveyed by the spout S. The teeth of the wheel F are so contrived as to receive one-half of the flat pistons, and let them fold in; and sometimes another wheel like F is fixed at the bottom D. The distance of the pistons from the side of the barrel is about half an inch; but as the machine is generally worked with great velocity, the ascending pistons bring along with them into the reservoir as much water as fills the cavity GH. Sometimes chain pumps are constructed without the barrels ABCD and GH. In this case, the flat pistons are converted into buckets connected with a chain, which dip in the water with their mouths downwards, and convey it to the reservoir. The buckets are moved by hexagonal axles, and the distance between each is nearly equal to the depth of the buckets. Chain pumps are frequently in an inclined position, and in this position they raise the greatest quantity of water when the distance of the flat pistons is equal to their breadth, and when the inclination of the barrels is about 24° 21'.

320. The hair-rope machine, invented by the Sieur Hair-rope Vera, operates on the same principle as the chain pump, machine of Instead of a chain of pistons moving round the wheel F, the Sieur a hair rope is substituted. The part of the rope at T Vera. that is lowest always dips in the water, which adhering Fig. 1. to the rope is raised along with it. When the rope reaches the top at G and M, it passes through two small tubes, which being fixed in the bottom of the re-

Sect. II. On Engines for Extinguishing Fire.

321. The common fire engine which discharges water in successive jets is represented in fig. 2, and is only a modification of the lifting pump. In the vessel AB full of water, is immersed the frame DC of a common lifting pump. This frame, and consequently the piston N, is elevated and depressed by means of the levers E, F, and the water which is raised is forced through the pipe G, which may be moved in any direction by means of the elastic leather pipe H, or by a ball and socket screwed on the top of the pump. While the piston N is descending, the stream at G is evidently discontinued, and issues only at each elevation of the piston. The vessel AB is supplied with water by buckets, and the pump is prevented from being choked by the strainer LK which separates from the water any mud that it may happen to contain.

322. As this fire engine does not afford a continued stream, it is not so useful in case of accidents as when the stream is uninterrupted. An improved engine of this sort is represented in fig. 3, where D, E, are two forcing pumps connected with the large vessel OG, and wrought by the levers F, G, moving upon H as a fulcrum. This apparatus is plunged and fastened in the vessel AB partly filled with water, and by means of the forcing pump DE, the operation of which has already been described, the water is driven through the valves I, L into the large vessel OG, where the included air is condensed. Into this vessel is inserted the tube PO communicating with the leathern pipe ORQS. The elasticity of the condensed air in the vessel OG pressing upon the surface of the water in that vessel, forces it up through the tube PO into the leathern pipe, from whose extremity S, it issues with great force and velocity; and as the condensed air is continually pressing upon the water in the vessel OG, the stream at S will be constant and uniform.

323. A section of the fire engine, as improved by Mr Newnham, is represented in fig. 4, where TU and WX are the forcing pumps corresponding with D and C in fig. 3. YZ the large vessel corresponding with GO, and ef the tube corresponding with PO. The vessels TU, WX, YZ, the horizontal canals ON, OP, ML, and the vertical canal EE, all communicate with each other by means of four valves O, I, K, P opening upwards, and the vertical pipe is immersed in the water to be raised. When the piston R is raised by means of the double lever ab, a vacuum would be made in the barrel TU, if the water at R were prevented from rising; but as this barrel communicates with the vessel of water below EF, on the surface of which the pressure of the atmosphere is exerted, the water will rise through EF, force open the valve H, and follow the piston R. By depressing the piston R, however, the water is driven down the barrel, closes the valve H, and rushes through the valve I into the air vessel YZ. The very same operation is going on with the pump WX, which forces the water into the air vessel through the valve K. By these means the air vessel is constantly filling with water, and the included air undergoing continual condensation. The air thus compressed, reacts upon the surface YZ of the water, and forces it through the tube ef to the stop-cock eg, whence, after turning the cock, the water passes into the tube h, fixed to a ball and socket, by which it may be discharged in any direction.

324. The fire engine has undergone various alterations Reference and improvements from Bramah, Dickenson, Simpkin, to the improvements Raventree, Philips and Furt, an account of whose engines may be seen in the Repertory of Arts, &c. A very simple and cheap fire engine has been invented by Mr B. Dearborn, and is described in the American Transactions for 1794, and in Gregory's Mechanics, vol. ii. p. 177.

Sect. III. On Whitehurst's Machine, and Montgolfier's Hydraulic Ram.

325. Mr Whitehurst * was the first who suggested the ingenious idea of raising water by means of its own momentum. A machine upon the same principle as Mr Whitehurst's, but in an improved form, has lately made its appearance in France, and excited considerable attention both on the continent and in this country. Whatever credit, therefore, has been given to the inventor of the hydraulic ram, justly belongs to our countryman Mr Whitehurst, and Montgolfier is entitled to nothing more than the merit of an improver.

326. Mr Whitehurst's machine, which was actually erected at Oulton in Cheshire, is represented in fig. 1, where AM is the original reservoir having its surface in the same horizontal line with the bottom of the reservoir BN. The diameter of the main pipe AE is one inch and a half, and its length about 200 yards; and the branch pipe EF is of such a size that the height of the surface M of the reservoir is nearly 16 feet above the cock F. In the valve box D is placed the valve e, and into the air vessel C are inserted the extremities m, n of the main pipe, bent downwards to prevent the air from being driven out, when the water is forced into it. Now as the cock F is 16 feet below the reservoir AM, the water will issue from F with a velocity of nearly 30 feet per second. As soon as the cock F therefore is opened, a column of water 200 yards long is put in motion, and though the aperture of the cock F be small, this column must have a very considerable momentum. Let the cock F be now suddenly stopped, and the water will rush through the valve d into the air vessel C, and condense the included air. This condensation must take place every time the cock is shut, and the imprisoned air being in a state of high compression, will react upon the water in the air vessel, and raise it into the reservoir BN.

327. A section of the hydraulic ram of Montgolfier Description is exhibited in fig. 2, where R is the reservoir, RS the height of the fall, and ST the horizontal canal which conveys the water to the engine ABHTC. E and D are two valves, and FG a pipe reaching within a very little of the bottom CB. Let us now suppose that wa- ter is permitted to descend from the reservoir. It will evidently rush out at the aperture m n till its velocity is such as to force up the valve E. The water being thus suddenly checked, and unable to find a passage at m n, will rush forwards to H and raise the valve D. A portion of water being thus admitted into the vessel ABC, the impulse of the column of fluid is spent, the valves D and E fall, and the water issues at m n as before; when its motion is again checked, and the same operation repeated, which has now been described. Whenever, therefore, the valve E closes, a portion of water will force its way into the vessel ABC, and condense the air which it contains, for the included air has no communication with the atmosphere after the bottom of the pipe FG is beneath the surface of the injected water. This condensed air will consequently react upon the surface of the water, and raise it in the pipe FG to an altitude proportioned to the elasticity of the included air. The external appearance of the engine, drawn from one in the possession of Professor Leslie, is represented in fig. 3, where ABC is the air vessel, F the valve box, G the extremity of the valve, and M, N screws for fixing the horizontal canal to the machine. When the engine is employed to form a jet of water, a piece of brafs, A, with a small aperture, is screwed upon the top of the tube FG, which, in that case, rises no higher than the top of the air vessel. From this description it will be seen, that the only difference between the engines of Montgolfier and Whitehurst, is, that the one requires a person to turn the cock, while the other has the advantage of acting spontaneously. Montgolfier (G) assures us, that the honour of this invention does not belong to England, but that he is the sole inventor, and did not receive a hint from any person whatever. We leave the reader to determine the degree of credit to which these assertions are entitled.—It would appear from some experiments made by Montgolfier, that the effect of the water ram is equal to between a half and three-fourths of the power expended, which renders it superior to most hydraulic machines.

Appendix to Ferguson's Lectures, p. 19.

SECT. IV. On Archimedes's Screw Engine.

328. THE screw engine invented by Archimedes is represented in fig. 4, where AB is a cylinder with a flexible pipe, CEHOGF, wrapped round its circumference like a forew. 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 water. When, by means of the handle K, the cylinder is made to revolve upon its axis, the water which enters 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 floatsboards 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 I. 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.

329. In order to explain the reason why the water rises in the spiral tube, let AB be a section of the engine, BC d DE 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 defend 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 move 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 e, the point d will be at C; and as the water at C cannot rise along with the point C to e, on account of the inclination of C c to the horizon, it must occupy the point d of the spiral, when C has moved to e; 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°.

330. The theory of this engine is treated at great length by Henrnet, in his Dissertation sur la vis d'Archimede, 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.

SECT. V. On the Persian Wheel.

331. THE Persian wheel is an engine which raises water to a height equal to its diameter. It is shewn in of the Persian wheel. Fig. 6, where CDE is the wheel driven by the stream AB acting upon floatboards fixed on one side of its rim. Fig. 6. A number of buckets, a, a, a, a, are disposed on the opposite side of the rim, and suspended by strong pins, b, b, b, &c. When the wheel is in motion, the descending buckets immerge into the stream, and ascend full

(G) Cette invention n'est point originaire d'Angleterre, elle appartient toute entière à la France. Je declare que j'en suis le seul inventeur, et que l'idée ne m'en a été fournie par personne. Journal des Mines, vol. xiii. No 73. full of water till they reach the top K, where they strike against the extremity n of the fixed reservoir M, and being overfull, discharge their contents into that reservoir. As soon as the bucket quits the reservoir, it resumes its perpendicular position by its own weight, and descends as before. On each bucket is fixed a spring r, which moves over the top of the bar m, fastened to the reservoir. By this means the bottom of the bucket is raised above the level of its mouth, and its contents completely discharged.

332. On some occasions the Persian wheel is made to raise water only to the height of its axle. In this case, instead of buckets, its spokes c, d, e, f, g, h, are made of a spiral form, and hollow within, so that their inner extremities all terminate in the box N on the axle, and their outer extremities in the circumference of the wheel. When the rim CDEF, therefore, is immersed in the stream, the water runs into the tubes C, D, E, F, &c. rises in the spiral spokes c, d, &c. and is discharged from the orifices at O into the reservoir Q, from which it may be conveyed in pipes.

SECT. VI. On the Zurich Machine.

333. This machine is a kind of pump invented and erected by H. Andreas Wirtz, an ingenious tin-plate worker in Zurich, and operates on a principle different from all other hydraulic engines. The following description of it, written by Dr Robison, is transferred to this part of the work for the sake of uniformity.

334. Fig. 7, is a sketch of the section of the machine, as it was first erected by Wirtz at a dye-house in Limmat, in the suburbs or vicinity of Zurich. It consists of a hollow cylinder, like a very large grindstone, turning on a horizontal axis, and partly plunged in a cistern of water. The axis is hollow at one end, and communicates with a perpendicular pipe CBZ, part of which is hid by the cylinder. This cylinder or drum is formed into a spiral canal by a plate coiled up within it like the main spring of a watch in its box; only the spires are at a distance from each other, so as to form a conduit for the water of uniform width. This spiral partition is well joined to the two ends of the cylinder, and no water escapes between them. The outermost turn of the spiral begins to widen about three-fourths of a circumference from the end, and this gradual enlargement continues from Q to S nearly a semicircle: this part may be called the Horn. It then widens suddenly, forming a Scoop or shovel SS'. The cylinder is supported so as to dip several inches into the water, whose surface is represented by VV'.

335. When this cylinder is turned round its axis in the direction ABEO, as expressed by the two darts, the scoop SS' dips at V' and takes up a certain quantity of water before it immerses again at V. This quantity is sufficient to fill the taper part SQ, which we have called the Horn; and this is nearly equal in capacity to the outermost uniform spiral round.

336. 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 mean time, air comes in at the mouth of the scoop; and when the scoop again dips into the water, it again takes in some. 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. Thus it will run over at K 4 into the right-hand side of the third spiral. Therefore it will push the water of this spire 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.

337. 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 mounts in the rising pipe; for the air next to the rising pipe is compressed at its inner end with the weight of the whole column in the main. It must be as much compressed at its outer end. This must be done by the water column without it; and this column exerts this pressure partly by reason that its outer end is higher than its inner end, and partly by the transmission of the pressure on its outer end by air, which is similarly compressed from without. And thus it will happen that each column of water, being higher at its outer than at its inner end, compresses the air on the water column beyond or within it, which transmits this pressure to the air beyond it, adding to it the pressure arising from its own want of level at the ends. Therefore the greatest compression, viz. that of the air next the main, is produced by the sum of all the transmitted pressures; and these are the sum of all the differences between the elevations of the inner ends of the water columns above their outer ends: and the height to which the water will rise in the main will be just equal to this sum.

338. Draw the horizontal lines K'K 1, K'K 2, K'K 3, &c. and mn, mn, mn, &c. Suppose the left-hand spaces to be filled with water, and the right-hand spaces to be filled with air. There is a certain gradation of compression which will keep things in this position. The spaces evidently decrease in arithmetical progression; so do the hydrostatic heights and pressures of the water columns. If therefore the air be dense in the same progression, all will be in hydrostatic equilibrium. Now this is evidently producible by the mere motion of the machine; for since the density and compression in each air column is supposed inversely as the bulk of the column, the absolute quantity of air is the same in all; therefore the column first taken in will pass gradually inwards, and the increasing compression will cause it to occupy precisely the whole right-hand side of every spire. The gradual diminution of the water columns will be produced during the motion by the water running over backwards at the top, from spire to spire, and at last coming out by the scoop.

339. It is evident that this disposition of the air and water will raise the water to the greatest height, because the hydrostatic height of each water column is the greatest possible, viz. the diameter of the spire. This disposition may be obtained in the following manner: Take CL to CB as the density of the external air to its density in the last column next the rising-pipe or main; that is, make CL to CB as 33 feet (the height of the column of water which balances the atmosphere), to the sum of 33 feet and the height of the rising-pipe. Then divide BL into such a number of turns, that the sum of their diameters shall be equal to the height of the main; then bring a pipe straight from L to the centre C. The reason of all this is very evident.

340. But when the main is very high, this construction will require a very great diameter of the drum, or many turns of a very narrow pipe. In such cases it will be much better to make the spiral in the form of a cork-crew, as in fig. 1, instead of this flat form like a watch spring. The pipe which forms the spiral may be lapped round the frustum of a cone, whose greatest diameter is to the least (which is next to the rising pipe) in the same proportion that we assigned to CB and CL. By this construction the water will stand in every round so as to have its upper and lower surfaces tangents to the top and bottom of the spiral, and the water columns will occupy the whole ascending side of the machine, while the air occupies the descending side.

341. This form is vastly preferable to the flat: it will allow us to employ many turns of a large pipe, and therefore produce a great elevation of a large quantity of water.

The same thing will be still better done by lapping the pipe on a cylinder, and making it taper to the end, in such a proportion that the contents of each round may be the same as when it is lapped round the cone. It will raise the water to a greater height (but with an increase of the impelling power) by the same number of turns, because the vertical or prefling height of each column is greater.

Nay, the same thing may be done in a more simple manner, by lapping a pipe of uniform bore round a cylinder. But this will require more turns, because the water columns will have less differences between the heights of their two ends. It requires a very minute investigation to show the progress of the columns of air and water in this construction, and the various changes of their arrangement, before one is attained which will continue during the working of the machine.

342. We have chosen for the description of the machine that construction which made its principle and manner of working most evident, namely, which contained the same material quantity of air in each turn of the spiral, more and more compressed as it approaches to the rising pipe. We should otherwise have been obliged to investigate in great detail the gradual progress of the water, and the frequent changes of its arrangement, before we could see that one arrangement would be produced which would remain constant during the working of the machine. But this is not the best construction. We see that, in order to raise water to the height of a column of 34 feet, which balances the atmosphere, the air in the last spire is compressed into half its bulk; and the quantity of water delivered into the main at each turn is but half of what was received into the first spire, the rest flowing back from spire to spire, and being discharged at the spout.

343. But it may be constructed so as that the quantity of water in each spire may be the same that was received into the first; by which means a greater quantity (double in the instance now given) will be delivered into the main, and raised to the same height by very nearly the same force.—This may be done by another proportion of the capacity of the spires, whether by a change of their caliber or of their diameters. Suppose the bore to be the same, the diameter must be made such that the constant column of water, and the column of air, compressed to the proper degree, may occupy the whole circumference. Let A be the column of water which balances the atmosphere, and h the height to which the water is to be raised. Let A be to \( \frac{A}{A+h} \) as 1 to m.

344. It is plain that m will represent the density of the air in the last spire, if its natural density be 1, because it is preflled by the column \( \frac{A}{A+h} \), while the common air is preflled by A. Let 1 represent the constant water column, and therefore nearly equal to the air column in the first spire. The whole circumference of the last spire must be \( 1 + \frac{1}{m} \), in order to hold the water 1, and the air compressed into the space \( \frac{1}{m} \) or \( \frac{A}{A+h} \).

345. The circumference of the first spire is \( 1 + 1 \) or 2. Let D and d be the diameters of the first and last spires; we have \( 2 : 1 + \frac{1}{m} = D : d \), or \( 2m : m+1 = D : d \). Therefore if a pipe of uniform bore be lapped round a cone, of which D and d are the end diameters, the spirals will be very nearly such as will answer the purpose. It will not be quite exact, for the intermediate spirals will be somewhat too large. The conoidal frustum should be formed by the revolution of a curve of the logarithmic kind. But the error is very trifling.

With such a spiral, the full quantity of water which was confined in the first spiral will find room in the last, and will be sent into the main at every turn. This is a very great advantage, especially when the water is to be much raised. The saving of power by this change of construction is always in proportion to the greatest compression of the air.

The great difficulty in the construction of any of these forms is in determining the form and position of the horn and the loop; and on this greatly depends the performance of the machine. The following instructions will make it pretty easy.

346. Let ABEO (fig. 2.) represent the first or outermost round of the spiral, of which the axis is C. Suppose it immersed up to the axis in the water VV, we have seen that the machine is most effective when the surfaces KB and On of the water columns are distant the whole diameter BO of the spiral. Therefore let the pipe be first supposed of equal calibre to the very mouth Ee, which we suppose to be just about to dip into the water. The surface On is kept there, in opposition to the pressure of the water column BAO, by the compressed air contained in the quadrant Oe, and in the quadrant which lies behind EB. And this compression is supported by the columns behind, between this spire and the rising pipe. But the air in the outermost quadrant EB is in its natural state, communicating as yet with the external air. When, however, the mouth Ee has come round to A, it will not have the water standing in it in the same manner, leaving the half space BEO filled with compressed air; for it took in and confined only what filled the quadrant B.E. It is plain, therefore, that the quadrant BE must be so shaped as to take in and confine a much greater quantity of air; so that when it has come to A, the space BEO may contain air sufficiently dense to support the column AO. But this is not enough: For when the wide mouth, now at A a, rises up to the top, the surface of the water in it rises also, because the part AO aa is more capacious than the cylindric part OEeo which succeeds it, and which cannot contain all the water that it does. Since, then, the water in the spire rises above A, it will press the water back from On to some other position m'n', and the pressing height of the water-column will be diminished by this rising on the other side of O. In short, the horn must begin to widen, not from B, but from A, and must occupy the whole semicircle ABE; and its capacity must be to the capacity of the opposite cylindrical side as the sum of BO, and the height of a column of water which balances the atmosphere to the height of that column. For then the air which filled it, when of the common density, will fill the uniform side BEO, when compressed so as to balance the vertical column BO. But even this is not enough; for it has not taken in enough of water. When it dipped into the cistern at E, it carried air down with it, and the pressure of the water in the cistern caused the water to rise into it a little way; and some water must have come over at B from the other side, which was drawing narrower. Therefore when the horn is in the position EOa, it is not full of water. Therefore when it comes into the situation OAB, it cannot be full nor balance the air on the opposite side. Some will therefore come out at O, and rise up through the water. The horn must therefore, 1st, Extend at least from O to B, or occupy half the circumference; and, 2dly, It must contain at least twice as much water as would fill the side BEO. It will do little harm though it be much larger; because the surplus of air which it takes in at E will be discharged, as the end Ee of the horn rises from O to B, and it will leave the precise quantity that is wanted. The overplus water will be discharged as the horn comes, round to dip again into the cistern. It is possible, but requires a diffusion too intricate for this place, to make it of such a size and shape, that while the mouth moves from E to B, passing through O and A, the surface of the water in it shall advance from E to O n, and be exactly at O when the beginning or narrow end of the horn arrives there.

347. We must also secure the proper quantity of water. When the machine is so much immered as to be up to the axis in water, the capacity which thus secures the proper quantity of air will also take in the proper quantity of water. But it may be erected so as that the spirals shall not even reach the water. In this case it will answer our purpose if we join to the end of the horn a scoop or shovel QRSB (fig. 3.), which is so formed as to take in at least as much water as will fill the horn. This is all that is wanted in the beginning of the motion along the spiral, and more than is necessary when the water has advanced to the succeeding spire; but the overplus is discharged in the way we have mentioned. At the same time, it is needless to load the machine with more water than is necessary, merely to throw it out again. We think that if the horn occupies fully more than one-half of the circumference, and contains as much as will fill the whole round, and if the scoop lifts as much as will certainly fill the horn, it will do very well.

N. B. The scoop must be very open on the side next the axis, that it may not confine the air as soon as it enters the water. This would hinder it from receiving water enough.

348. The following dimensions of a machine erected at Florence, and whose performance corresponded extremely well with the theory, may serve as an example.

The spiral is formed on a cylinder of 10 feet diameter, and the diameter of the pipe is six inches. The smaller end of the horn is of the same diameter; it occupies three-fourths of the circumference, and is 7 9/10ths inches wide at the outer end. Here it joins the scoop, which lifts as much water as fills the horn, which contains 4340 Swedish cubic inches, each = 1.577 English. The machine makes six turns in a minute, and raises 1354 pounds of water, or 22 cubic feet, 10 feet high in a minute.

349. The above account will, we hope, sufficiently explain the manner in which this singular hydraulic machine produces its effect. When every thing is executed by the maxims which we have deduced from its principles, we are confident that its performance will correspond to the theory; and we have the Florentine machine as a proof of this. It raises more than ten-elevenths of what the theory promises, and it is not perfect. The spiral is of equal caliber, and is formed on a cylinder. The friction is so inconsiderable in this machine, that it need not be minded: but the great excellency is, that whatever imperfection there may be in the arrangement of the air and water columns, this only affects the elegance of the execution, causing the water to make a few more turns in the spiral before it can mount to the height required; but wastes no power, because the power employed is always in proportion to the sum of the vertical columns of water in the rising side of the machine; and the height to which the water is raised by it is in the very same proportion. It should be made to move very slow, that the water be not always dragged up by the pipes, which would cause more to run over from each column, and diminish the pressure of the remainder.

350. If the rising-pipe be made wide, and thus room be made for the air to escape freely up through the water, it will rise to the height assigned; but if it be narrow, so that the air cannot get up, it rises almost as slow as the water, and by this circumstance the water is raised to a much greater height mixed with air, and this with hardly any more power. It is in this way that we can account for the great performance of the Florentine machine, which is almost triple of what a man can do with the finest pump that ever was made: indeed the performance is so great, that one is apt to suspect some inaccuracy in the accounts. The entry into the rising-pipe should be no wider than the last part of the spiral; and it would be advisable to divide it into four channels by a thin partition, and then to make the rising-pipe very wide, and to put into it a number of slender rods, which would divide it into slender channels that would completely entangle the air among the water. This will greatly increase the height of the heterogeneous column. It is surprising that a machine that is so very promising should have attracted so little notice. We do not know of any being erected out of Switzerland, except at Florence in 1778. The account of its performance was in consequence of a very public trial in 1779, and honourable declaration of its merit, by Sig. Lorenzo Ginori, who erected another, which fully equalled it. It is shortly mentioned by Professor Sulzer of Berlin, in the Sammlungen Vermischten Schriften for 1754. A description of it is published by the Philosophical Society at Zurich in 1766, and in the descriptions published by the Society in London for the encouragement of Arts in 1776. The celebrated Daniel Bernoulli has published a very accurate theory of it in the Peterburgh Commentaries for 1772, and the machines at Florence were erected according to his instructions. Baron Aldromer in Sweden caused a glass model of it to be made, to exhibit the internal motions for the instruction of artists, and also ordered an operative engine to be erected; but we have not seen any account of its performance. It is a very intricate machine in its principles; and an ignorant engineer, nay the most intelligent, may erect one which shall hardly do any thing; and yet by a very trifling change, may become very powerful. We presume that failures of this kind have turned the attention of engineers from it; but we are persuaded that it may be made very effective, and we are certain that it must be very durable.

Fig. 4. is a section of the manner in which the author has formed the communication between the spiral and the rising-pipe. P is the end of the hollow axis which is united with the solid iron axis. Adjoining to P, on the under side, is the entry from the last turn of the spiral. At Q is the collar which rests on the supports, and turns round in a hole of bell-metal. ff is a broad flanch cast in one piece with the hollow part. Beyond this the pipe is turned somewhat smaller, very round and smooth, so as to fit into the mouth of the rising-pipe, like the key of a cock. This mouth has a plate ee attached to it. There is another plate dd, which is broader than ee, and is not fixed to the cylindrical part, but moves easily round it. In this plate are four screws, such as g, g, which go into holes in the plate ff, and thus draw the two plates ff and dd together, with the plate ee between them. Pieces of thin lead-plate are put on each side of ee; and thus all escape of water is effectually prevented, with a very moderate compression and friction.

CHAP. IV. On Machines in which Water is the chief Agent.

SECT. I. On the Water Blowing Machine.

351. The water blowing machine consists of a reservoir of water AB, into the bottom of which the bent leaden pipe BCH is inserted; of a condensing vessel DE, into whose top the lower extremity H of the pipe is fixed, and of a pedestal P resting on the bottom of this vessel. When the water from the reservoir AB is descending through the part CH of the pipe, it is in contact with the external air by means of the orifices or tubes m, n, o, p; and by the principle of the lateral communication of motion in fluids (art. 160.), the air is dragged along with the water. This combination of air and water issuing from the aperture H, and impinging upon the surface of the stone pedestal P, is dispersed in various directions. The air being thus separated from the water, ascends into the upper part of the vessel, and rushes through the opening F, whence it is conveyed by the pipe FG to the fire at G, while the water falls to the lower part of the vessel, and is discharged by the openings M, N.—That the greatest quantity of air may be driven into the vessel DE, the water should begin to fall at C with the least possible velocity; and the height of the lowest tubes above the extremity H of the pipe should be three-elevenths of the length of the vertical tube CH, in order that the air may move in the pipe FG with sufficient velocity.

352. Fabri and Dietrich imagined that the wind is produced by the decomposition of the water, or its transformation into gas, in consequence of the agitation and percussion of its parts. But M. Venturi, to whom we owe the first philosophical account of this machine, has shewn that this opinion is erroneous, and that the wind is supplied from the atmosphere, for no wind was generated when the lateral openings m, n, o, p were shut. The principal object, therefore, in the construction of water blowing machines, is to combine as much air as possible with the descending current. For this purpose the water is often made to pass through a kind of culender placed in the open air, and perforated with a number of small triangular orifices. Through these apertures the water descends in many small streams; and by exposing a greater surface to the atmosphere, it carries along with it an immense quantity of air. The water is then conveyed to the pedestal P by a pipe CH opened and enlarged at C, so as to be considerably wider than the end of the tube which holds the culender.

353. It has been generally supposed that the waterfall should be very high; but Dr Lewis has shewn, by a variety of experiments, that a fall of four or five feet is sufficient, and that when the height is greater than this, two or more blowing machines may be erected, by conducting the water from which the air is extricated, into another reservoir, from which it again descends, and Bramah's generates air as formerly. In order that the air which is necessarily loaded with moisture, may arrive at the furnace in as dry a state as possible, the condensing vessel DE should be made as high as circumstances will permit; and in order to determine the strength of the blast, it should be furnished with a gage ab filled with water.

354. The rain wind is produced in the same way as the blast of air in water blowing machines. When the drops of rain impinge upon the surface of the sea, the air which they drag along with them often produces a heavy squall, which is sufficiently strong to carry away the mast of a ship. The same phenomenon happens at land, when the clouds empty themselves in alternate showers. In this case, the wind proceeds from that quarter of the horizon where the shower is falling. The common method of accounting for the origin of the winds by local rarefaction of the air appears pregnant with insuperable difficulties; and there is reason to think that these agitations in our atmosphere ought rather to be referred to the principle which we have now been considering. For farther information on this subject, the reader is referred to Lewis's Commerce of Arts, Wolfi Opera Mathematica, tom. i. p. 830. Journal des Mines, No xci, or Nicholson's Journal, vol. xii. p. 48.

Sect. II. Bramah's Hydrostatic Prefs.

355. The machine invented by Mr Bramah of Piccadilly, depends upon the principle, that any pressure exerted upon a fluid mass is propagated equally in every direction (art. III.). It is represented in fig. 6, where A is a strong metallic cylinder, furnished with a piston B perfectly water-tight. Into the bottom of this cylinder is inserted the end of the bent tube C, the interior orifice of which is closed by the valve D. The other extremity of the tube communicates with the forcing pump E, by which water or other fluids may be driven into the cylinder A. Then, if any pressure is exerted on the surface of the water in the cylinder E, by means of the lever H, this pressure will be propagated to the cylinder A, and exert a certain force upon the piston B, varying with the respective areas of the sections of each cylinder. If the diameter of the cylinder E is equal to the diameter of the cylinder A, and if a force of 10 pounds is exerted at the handle H, then the piston B will be elevated with a force of 10 pounds; if the diameter of E be one-half that of A, the piston B will be raised with a force of 40 pounds, because the area of the one piston is four times the area of the other. Or, in general, if D be the diameter of the cylinder A, d that of the cylinder E, and F the force exerted at the lever H, we shall have \( d^2 : D^2 = F : \frac{F \times D^2}{d^2} \), which is the force exerted upon the piston B.

Thus, if \( d = 2 \) inches, \( D = 24 \) inches, and \( F = 10 \) pounds, then \( \frac{F \times D^2}{d^2} = \frac{10 \times 24 \times 24}{2 \times 2} = 1440 \) pounds, the force with which the piston B is elevated. Now, as this force increases as \( d^2 \) diminishes, or as F and \( D^2 \) increase, there is no limit to the power of the engine; for the diameter of the cylinder A may be made of any size, and that of the cylinder E exceedingly small, while the power may be still farther augmented by lengthening the lever H. The same effects may be produced by injecting air into the pipe C by means of a large globe fixed at its extremity. Upon the same principles the power and motion of one machine may be communicated to another; for we have only to connect the two machines by means of a pipe filled with water, inserted at each extremity into a cylinder furnished with a piston. By this means the power which deprestes one of the pistons will be transferred along the connecting pipe, and will elevate the other piston. In the same way water may be raised out of wells of any depth, and at any distance from the place where the power is applied; but we must refer the reader, for a detailed account of these applications, to the specification of the patent obtained by Mr Bramah, or to Gregory's Mechanics, vol. ii. p. 120.

Sect. III. On Clepsydrae or Water-Clocks.

356. A clepsydra or water-clock, derived from History of κλεπτής, "to steal," and ὕδωρ, "water," is a machine clepsydra, which measures time by the motion of water (art. 159.) The invention of this machine has been ascribed to Scipio Nafica, the cousin of Scipio Africanus, who flourished about 200 years before the Christian era. It was well known, however, at an earlier period, among the Egyptians, who employed it to measure the course of the sun. It is highly probable that Scipio Nafica had only the merit of introducing it into his native country. These machines were in use for a very long period, and continued to be employed as measurers of time till the invention of the pendulum clock enriched the arts and sciences.

357. The clepsydra, invented by Ctesibius of Alexandria, was an interesting machine. The water which indicated the progress of time by the gradual descent of its surface, flowed in the form of tears from the eyes of a human figure. Its head was bent down with age: its look was dejected, while it seemed to pay the last tribute of regret to the fleeting moments as they passed. —The water which was thus discharged was collected in a vertical reservoir, where it raised another figure holding in its hand a rod, which, by its gradual ascent, pointed out the hours upon a vertical column. The same fluid was afterwards employed in the interior of the pedestal, as the impelling power of a piece of machinery which made this column revolve round its axis in a year, so that the months and the days were always shewn by this index, whose extremity described a vertical line divided according to the relative lengths of the hours of day and night. Among the ancients the length of the hours varied every day, and even the hours of the day differed in length from those of the night; for the length of the day, or the interval between sunrise and sunset, was always divided into twelve equal parts, while the length of the night, or the interval between sunset and sunrise, was divided into the same number of parts, for hours. A farther description of this beautiful machine, and others of the same nature, may be seen in Perrault's Vitruvius.

358. The method of constructing clepsydrae, when the vessel from which the fluid issues is cylindrical or of any other form, has been shewn in Prop. VII. Part II. Instead of dividing the sides of the vessel, for a scale to ascertain the descent of the fluid surface, the following

The following method may be adopted. In the bottom of the cylindrical vessel ABCD, which is about 12 inches high, and four inches in diameter, is inserted a small glass adjutant E, which discharges the water in the vessel by successive drops. A hole F, about half an inch in diameter, is perforated in the cover AB, so as to allow the glass tube GI, about 16 inches long, and half an inch in diameter, to move up and down without experiencing any resistance. To the extremity of this tube is attached the ball I, which floats on the surface of the water in the vessel, and is kept steady, either by introducing a quantity of mercury into its cavity, if it be hollow, or by suspending a weight if it is a solid which does not sink in water. When the vessel is filled with water, the ball I will be at the top AB; then, in order to graduate the tube C, let the water flow out at E, and by means of a watch mark the points on the tube which descend to F after the lapse of every hour, every half hour, and every quarter, and the instrument will be finished. In order to use this hydroscope or water-clocks, pour water into the vessel ABCD till the hour of the day is about to descend below F; and when this is done, it will point out any succeeding hour till the vessel is emptied.

359. The clepsydra, invented by the honourable Mr Charles Hamilton, is represented in fig. 7. An open canal ee, supplied with a constant and equal stream by the syphon d, has at each end ff, open pipes f1, f2 of exactly equal bores, which deliver the water that runs along the canal e, alternately into the vessels g 1, g 2, in such a quantity as to raise the water from the mouth of the tantalus t, exactly in an hour. The canal ee is equally poised by the two pipes f1, f2, upon a centre r; the ends of the canal e are raised alternately, as the cups zx are depressed, to which they are connected by lines running over the pulleys ll. The cups zx are fixed at each end of the balance mm, which moves up and down upon its centre v. n 1, n 2, are the edges of two wheels or pulleys, moving different ways alternately, and fitted to the cylinder o by oblique teeth both in the cavity of the wheel and upon the cylinder, which, when the wheel z moves one way, that is, in the direction of the minute-hand, meet the teeth of the cylinder and carry the cylinder along with it, and slip over those of the cylinder when z moves the contrary way, the teeth not meeting, but receding from each other. One or other of these wheels nn continually moves o in the same direction, with an equable and uninterrupted motion. A fine chain goes twice round each wheel, having at one end a weight X, always out of the water, which equiponderates with y at the other end, when kept floating on the surface of the fluid in the vessel g, which y must always be; the two cups zx, one at each end of the balance, keep it in equilibrium, till one of them is forced down by the weight and impulse of the water, which it receives from the tantalus t ii. Each of these cups zx, zx, has likewise a tantalus of its own h, h, which empties it after the water has run from g, and leaves the two cups again in equilibrium: q is a drain to carry off the water. The dial-plate, &c. needs no description. The motion of the clepsydra is effected thus: As the end of the canal ee, fixed to the pipe f1, is the lowest in the figure, all the water supplied by the syphon runs through the pipe f1, into the vessel g 1, till it runs over the top of the tantalus t; when it immediately runs out at t into the cup Z, at the end of the balance m, and forces it down; the balance moving on its centre v. When one side of m is brought down, the string which connects it to f1, running over the pulley l, raises the end f1, of the canal e, which turns upon its centre r, higher than f2; consequently, all the water which runs through the syphon d passes through f2 into g 2, till the same operation is performed in that vessel, and so on alternately. As the height to which the water rises in g in an hour, viz. from S to r, is equal to the circumference of v, the float y rising through that height along with the water, allows the weight X to act upon the pulley n, which carries with it the cylinder o; and this, making a revolution, causes the index k to describe an hour on the dial-plate. This revolution is performed by the pulley n 1; the next is performed by n 2, whilst n 1 goes back, as the water in g 1 runs out through the tantalus; for y must follow the water, as its weight increases, out of it. The axis o always keeps moving the same way; the index p describes the minutes; each tantalus must be wider than the syphon, that the vessels gg may be emptied as low as r, before the water returns to them.

360. For farther information respecting subjects connected with hydrodynamics, see the articles FLOATING Bodies, MECHANICS, MILL, PUMP, RESISTANCE of Fluids, RIVER, SPECIFIC Gravity, SHIP-Building, and WATER Works.