a well known fluid, diffused through the atmosphere, and over the surface of the globe, and abounding in a certain proportion in animals, vegetables, and minerals.

The uses of water are so universally known, that it would be superfluous to enumerate them in this article. It is essential to animal and vegetable life; it makes easy the intercourse between the most distant regions of the world; and it is one of the most useful powers in the mechanic arts. It is often found combined with various substances, and is then frequently beneficial in curing or alleviating diseases.

These properties of water which fit it for answering mechanical purposes are explained in other articles of this work (see Hydrodynamics, Pneumatics, No 3, Water Resistance, and Rivers); and for the discovery of the composition of water, see Chemistry Index.

Mineral Waters. For the method of analysing them, also see Chemistry Index.

Under the title of Mineral Waters, we have given an analysis of the most remarkable waters in Europe.

Holy Water, which is made use of in the church of Rome, as also by the Greeks, and by the other Christians of the East of all denominations, is water with a mixture of salt, blessed by a priest according to a set form of benediction. It is used in the blessing of persons, things, and places; and is likewise considered as a ceremony to excite pious thoughts in the minds of the faithful.

The priests, in blessing it, first, in the name of God, commands the devils not to hurt the persons who shall be sprinkled with it, nor to abuse the things, nor disquiet the places, which shall likewise be so sprinkled. He then prays that health, safety, and the favour of heaven, may be enjoyed by such persons, and by those who shall use such things, or dwell in such places. Vestments, vessels, and other such things that are set apart for divine service, are sprinkled with it. It is sometimes sprinkled on cattle, with an intention to free or preserve them from diabolical enchantments; and in some spiritual books there are prayers to be said on such occasions, by which the safety of such animals, as being a temporal blessing to the possessors, is begged of God, whose providential care is extended to all his creatures. The hope which Catholics entertain of obtaining such good effects from the devout use of holy water, is grounded on the promise made to believers by Christ (St Mark xvi. 17); and on the general efficacy of the prayers of the church; the petition of which prayers God is often pleased to grant; though sometimes, in his Providence, he sees it not expedient to do so. That such effects have been produced by holy water in a remarkable manner, has been asserted by many authors of no small weight; as, namely, by St Epiphanius, Haer. 30th; St Hieron., in the Life of St Hilarion; Theodore Hist. Eccl. lib. v. cap. 21.; Palladius, Hist. Laus.; Bede, lib. v. cap. 4.

As a ceremony (says the Catholic), water brings to our remembrance our baptism; in which, by water, we were cleansed from original sin. It also puts us in mind of that purity of conscience which we ought to endeavour always to have, but especially when we are going to worship our God. The salt, which is put into the water to preserve it from corrupting, is also a figure of divine grace, which preserves our souls from the corruption of sin; and is likewise an emblem of that wisdom and discretion which ought to season every action that a Christian does, and every word that he says. It is wont to be blessed and sprinkled in churches on Sundays, in the beginning of the solemn office. It is kept in vessels at the doors of the same churches, that it may be taken by the faithful as they enter in. It is also often kept in private houses and chambers.

Putrid Water, is that which has acquired an offensive smell and taste by the putrescence of animal or vegetable substances contained in it. It is in the highest degree pernicious to the human frame, and capable of bringing on mortal diseases even by its smell. It is not always from the apparent muddiness of waters that we Water can judge of their disposition to putrefy; some which are seemingly very pure, being more apt to become putrid than others which appear much more mixed with heterogeneous matters. Under the article Animalcule, No. 33, is mentioned a species of insects which have the property of making water stink to an incredible degree, though their bulk in proportion to the fluid which surrounds them is less than that of one to a million. Other substances no doubt there are which have the same property; and hence almost all water which is confined from the air is apt to become offensive, even though kept in glass or stoneware vessels. Indeed it is a common observation, that water keeps much longer sweet in glass vessels, or in those of earthen or stoneware than in those of wood, where it is exceedingly apt to putrefy. Hence, as ships can only be supplied with water kept in wooden casks, sailors are extremely liable to those diseases which arise from putrid water; and the discovery of a method by which water could easily be prevented from becoming putrid at sea would be exceedingly valuable. This may indeed be done by quicklime; for when water is impregnated with it, all putrescent matters are either totally destroyed, or altered in such a manner as never to be capable of undergoing the putrefactive fermentation again. But a continued use of lime-water could not fail of being pernicious, and it is therefore necessary to throw down the lime; after which the water will have all the purity necessary for preserving it free from putrefaction. This can only be done by means of fixed air; and mere exposure in broad shallow vessels to the atmosphere would do it without anything else, only taking care to break the crust which formed upon it.

Two methods, however, have been thought of for doing this with more expedition. The one, invented by Dr Alston, is, by throwing into the water impregnated with lime a quantity of magnesia. The lime attracts fixed air more powerfully than magnesia; in consequence of which the latter parts with it to the lime; and thus becoming insoluble, falls along with the caustic magnesia to the bottom, and thus leaves the water perfectly pure. Another method is that of Mr Henry, who proposes to throw down the lime by means of an effervescent mixture of oil of vitriol and chalk put down to the bottom of the water cask. His apparatus for this purpose is as simple as it can well be made, though it is hardly probable that sailors will give themselves the trouble of using it; and Dr Alston's scheme would seem better calculated for them, were it not for the expense of the magnesia; which indeed is the only objection made to it by Mr Henry. Putrid water may be restored and made potable by a process of the same kind.

Of late it has been discovered that charcoal possesses many unexpected properties, and, among others, that of preserving water from corruption, and of purifying it after it has been corrupted. Mr Lowitz, whose experiments on charcoal have been published in Crell's Chemical Journal, has turned his attention to this subject in a memoir read to the Economical Society at Petersburgh. He found that the effect of charcoal was rendered much more speedy by using along with it some sulphuric acid. One ounce and a half of charcoal in powder, and 24 drops of concentrated sulphuric acid (oil of vitriol), are sufficient to purify three pints and a half of corrupted water, and do not communicate to it any sensible acidity. This small quantity of acid renders it unnecessary to use more than a third part of the charcoal powder which would otherwise be wanted; and the less of that powder is employed, the less is the quantity of water lost by the operation, which, in sea-voyages, is an object worthy of consideration. In proportion to the quantity of acid made use of, the quantity of charcoal may be diminished or augmented. All acids produce nearly the same effects; neutral salts also, particularly nitre and sea-salt, may be used, but sulphuric acid is preferable to any of these; water which is purified by means of this acid and charcoal will keep a longer time than that which is purified by charcoal alone. When we mean to purify any given quantity of corrupted water, we should begin by adding to it as much powder of charcoal as is necessary to deprive it entirely of its bad smell. To ascertain whether that quantity of powdered charcoal was sufficient to effect the clarification of the said water, a small quantity of it may be passed through a linen bag, two or three inches long; if the water, thus filtrated, still has a turbid appearance, a fresh quantity of powdered charcoal must be added, till it become perfectly clear: the whole of the water may then be passed through a filtering bag, the size of which should be proportioned to the quantity of water. If sulphuric acid, or any other, can be procured, a small quantity of it should be added to the water, before the charcoal powder.

The cleaning of the casks in which water is to be kept in sea-voyages should never be neglected: they should be well washed with hot water and sand, or with any other substance capable of removing the mucilaginous particles, and afterwards a quantity of charcoal dust should be employed, which will entirely deprive them of the musty or putrid smell they may have contracted.—The charcoal used for purifying water should be well burnt, and afterwards beat into a fine powder.

Sea-Water. See Sea-Water.

Water-Carts, carriages constructed for the purpose of watering the roads for several miles round London; a precaution absolutely necessary near the metropolis, where, from such a vast daily influx of carriages and horses, the dust would otherwise become quite insufferable in hot dry weather. Pumps are placed at proper distances to supply these carts.

Water-Ordeal. See Ordeal.

Water, among jewellers, is properly the colour or lustre of diamonds and pearls. The term, though less properly, is sometimes used for the hue or colour of other stones.

Water-Bellows. See Machines for blowing Air into Furnaces.

Water-Colours, in Painting, are such colours as are only diluted and mixed up with gum-water, in contradistinction to oil-colours. See Colour-Making.

Water-Gang, a channel cut to drain a place by carrying off a stream of water.

Water-Hen. See Parra, Ornithology Index.

Water-Lines of a Ship, certain horizontal lines supposed to be drawn about the outside of a ship's bottom, close to the surface of the water in which she floats. They are accordingly higher or lower upon the bottom, in proportion to the depth of the column of water required to float her.

Water-Logged, the state of a ship when, by receiving a great quantity of water into the hold, by leaking, &c., she she has become heavy and inactive upon the sea, so as to yield without resistance to the efforts of every wave rushing over her decks. As, in this dangerous situation, the centre of gravity is no longer fixed, but fluctuating from place to place, the stability of the ship is utterly lost; she is therefore almost totally deprived of the use of her sails, which would operate to overturn her, or press the head under water. Hence there is no resource for the crew, except to free her by the pumps, or to abandon her by the boats as soon as possible.

**Water-Sail**, a small sail spread occasionally under the lower studding-sail, or driver boom, in a fair wind and smooth sea.

**Water-Ouzel.** See **Turdus, Ornithology Index.**

**Water-Spout,** an extraordinary meteor, consisting of a large mass of water collected into a sort of column, and moved with rapidity along the surface of the sea.

The best account of the water-spout which we have met with is in the Phil. Trans. Abridged, vol. iii. as observed by Mr Joseph Harris, May 21, 1732, about sunset, lat. 32° 30' N.; long. 9° E. from Cape Florida.

"When first we saw the spout (says he), it was whole and entire, and much of the shape and proportion of a speaking trumpet; the small end being downwards, and reaching to the sea, and the big end terminated in a black thick cloud. The spout itself was very black, and the more so the higher up. It seemed to be exactly perpendicular to the horizon, and its sides perfectly smooth, without the least ruggedness. Where it fell the spray of the sea rose to a considerable height, which made somewhat the appearance of a great smoke. From the first time we saw it, it continued whole about a minute, and till it was quite dissipated about three minutes. It began to waste from below, and so gradually up, while the upper part remained entire, without any visible alteration, till at last it ended in the black cloud above; upon which there seemed to fall a very heavy rain in that neighbourhood.—There was but little wind, and the sky elsewhere was pretty serene."

Water-spouts have by some been supposed to be merely electrical in their origin; particularly by Signior Beccaria, who supported his opinion by some experiments. But if we attend to the successive phenomena necessary to constitute a complete water-spout through their various stages, we shall be convinced, that recourse must be had to some other principle in order to obtain a complete solution.

Dr Franklin, in his Physical and Meteorological Observations, supposes a water-spout and a whirlwind to proceed from the same cause; their only difference being, that the latter passes over the land, and the former over the water. This opinion is corroborated by M. de la Pryme, in the Philosophical Transactions, where he describes two spouts observed at different times in Yorkshire, whose appearances in the air were exactly like those of the spouts at sea, and their effects the same as those of real whirlwinds.

A fluid moving from all points horizontally towards a centre, must at that centre either mount or descend. If a hole be opened in the middle of the bottom of a tub filled with water, the water will flow from all sides to the centre, and there descend in a whirl; but air flowing on or near the surface of land or water, from all sides towards a centre, must at that centre ascend; because the land or water will hinder its descent.

Vol. XX. Part II.

The doctor, in proceeding to explain his conceptions, begs to be allowed two or three positions, as a foundation for his hypothesis. 1. That the lower region of air is often more heated, and so more rarefied, than the upper, and by consequence specifically lighter. The coldness of the upper region is manifested by the hail, which falls from it in warm weather. 2. That heated air may be very moist, and yet the moisture so equally diffused, and rarefied as not to be visible till colder air mixes with it; at which time it condenses and becomes visible. Thus our breath, although invisible in summer, becomes visible in winter.

These circumstances being granted, he presupposes a tract of land or sea, of about 60 miles in extent, unsheltered by clouds and unrefreshed by the wind, during a summer's day, or perhaps for several days without intermission, till it becomes violently heated, together with the lower region of the air in contact with it; so that the latter becomes specifically lighter than the superincumbent higher region of the atmosphere, wherein the clouds are usually floated; he supposes also that the air surrounding this tract has not been so much heated during those days, and therefore remains heavier. The consequence of this, he conceives, should be, that the heated lighted air should ascend, and the heavier descend; and as this rising cannot operate throughout the whole tract at once, because that would leave too extensive a vacuum, the rising will begin precisely in that column which happens to be lightest or most rarefied; and the warm air will flow horizontally from all parts of this column, where the several currents meeting, and joining to rise, a whirl is naturally formed, in the same manner as a whirl is formed in a tub of water, by the descending fluid receding from all sides of the tub towards the hole in the centre.

And as the several currents arrive at this central rising column with a considerable degree of horizontal motion, they cannot suddenly change it to a vertical motion; therefore as they gradually, in approaching the whirl, decline from right to curve or circular lines, so, having joined the whirl, they ascend by a spiral motion; in the same manner as the water descends spirally through the hole in the tub before mentioned.

Lastly, as the lower air nearest the surface is more rarefied by the heat of the sun, it is more impressed by the current of the surrounding cold and heavy air which is to assume its place, and consequently its motion towards the whirl is swiftest, and so the force of the lower part of the whirl strongest, and the centrifugal force of its particles greatest. Hence the vacuum which incloses the axis of the whirl should be greatest near the earth or sea, and diminish gradually as it approaches the region of the clouds, till it ends in a point.

This circle is of various diameters, sometimes very large.

If the vacuum passes over water, the water may rise in a body or column therein to the height of about 32 feet. The whirl of air may be as invisible as the air itself, though reaching in reality from the water to the region of cool air, in which our low summer thunderclouds commonly float; but it will soon become visible at its extremities. The agitation of the water under the whirling of the circle, and the swelling and rising of the water in the commencement of the vacuum, render it visible below. It is perceived above by the

† 4 N warm warmed air being brought up to the cooler region, where its moisture begins to be condensed by the cold into thick vapour, and is then first discovered at the highest part, which being now cooled condenses what rises behind it, and this latter acts in the same manner on the succeeding body; where, by the contact of the vapours, the cold operates faster in a right line downwards, than the vapours themselves can climb in a spiral line upwards: they climb however; and as by continual addition they grow denser, and by consequence increase their centrifugal force, and being risen above the concentrating currents that compose the whirl, they fly off, and form a cloud.

It seems easy to conceive, how by this successive condensation from above, the sport appears to drop or descend from the cloud, although the materials of which it is composed are all the while ascending. The condensation of the moisture contained in so great a quantity of warm air as may be supposed to rise in a short time in this prodigiously rapid whirl, is perhaps sufficient to form a great extent of cloud; and the friction of the whirling air on the sides of the column may detach great quantities of its water, disperse them into drops, and carry them up in the spiral whirl mixed with the air. The heavier drops may indeed fly off, and fall into a shower about the spout; but much of it will be broken into vapour, and yet remain visible.

As the whirl weakens, the tube may apparently separate in the middle; the column of water subsiding, the superior condensed part drawing up to the cloud. The tube or whirl of air may nevertheless remain entire, the middle only becoming invisible, as not containing any visible matter.

Dr Lindsay, however, in several letters published in the Gentleman's Magazine, has controverted this theory of Dr Franklin, and endeavoured to prove that waterspouts and whirlwinds are distinct phenomena; and that the water which forms the water-spout, does not ascend from the sea, as Dr Franklin supposes, but descends from the atmosphere. Our limits do not permit us to insert his arguments here, but they may be seen in the Gentleman's Magazine, volume lii. p. 559; vol. liii. p. 1025; and vol. lv. p. 594. We cannot avoid observing, however, that he treats Dr Franklin with a degree of asperity to which he is by no means entitled, and that his arguments, even if conclusive, prove nothing more than that some water-spouts certainly descend; which Dr Franklin hardly ever ventured to deny. There are some very valuable dissertations on this subject by Professor Wilcke of Upsal.

WATER-WORKS.

Under this name may be comprehended almost every hydraulic structure or contrivance; such as canals, conduits, locks, mills, water engines, &c. But they may be conveniently arranged under two general heads, 1st, Works which have for their object the conducting, raising, or otherwise managing, of water; and, 2dly, Works which derive their efficacy from the impulse or other action of water. The first class comprehends the methods of simply conducting water in aqueducts or in pipes for the supply of domestic consumption or the working of machinery: It comprehends also the methods of procuring the supplies necessary for these purposes, by means of pumps, water, or fire engines. It also comprehends the subsequent management of the water thus conducted, whether in order to make the proper distribution of it according to the demand, or to employ it for the purpose of navigation, by lockage, or other contrivances.—And in the prosecution of these things many subordinate problems will occur, in which practice will derive great advantages from a scientific acquaintance with the subject. The second class of water-works is of much greater variety, comprehending almost every kind of hydraulic machine; and would of itself fill volumes. Many of these have already occurred in various articles of this Dictionary. In describing or treating them, we have tacitly referred the discussion of their general principles, in which they all resemble each other, to some article where they could be taken in a connected body, susceptible of general scientific discussion, independent of the circumstances which of necessity introduced the particular modifications required by the uses to which the structures were to be applied. That part of the present article, therefore, which embraces these common principles, will chiefly relate to the theory of water mills, or rather of water wheels; because, when the necessary motion is given to the axis of the water wheel, this may be set to the performance of any task whatever.

CLASS I.

1. Of the Conducting of Water.

This is undoubtedly a business of great importance, and makes a principal part of the practice of the civil engineer: It is also a business so imperfectly understood, that we believe that very few engineers can venture to say, with tolerable precision, what will be the quantity of water which his work will convey, or what plan and dimensions of conduit will convey the quantity which may be proposed. For proof of this we shall only refer our readers to the facts mentioned in the article Rivers, No 27, &c.

In that article we have given a sort of history of the progress of our knowledge in hydraulics, a branch of mechanical philosophy which seems to have been entirely unknown to the ancients. Even Archimedes, the author of almost all that we know in hydrostatics, seems to have been entirely ignorant of any principles by which he could determine the motion of water. The mechanical science of the ancients seems to have reached no farther than the doctrine of equilibrium among bodies at rest. Guglielmini first ventured to consider the motion of water in open canals and in rivers. Its motion in pipes had been partially considered in detached tached scraps by others, but not so as to make a body of doctrine. Sir Isaac Newton first endeavoured to render hydraulics susceptible of mathematical demonstration: But his fundamental proposition has not yet been freed from very serious objections; nor have the attempts of his successors, such as the Bernoullis, Euler, D'Alembert, and others, been much more successful: so that hydraulics may still be considered as very imperfect, and the general conclusions which we are accustomed to receive as fundamental propositions are not much better than matters of observation, little supported by principle, and therefore requiring the most scrupulous caution in the application of them to any hitherto untried case. When experiments are multiplied so as to include as great a variety of cases as possible; and when these are cleared of extraneous circumstances, and properly arranged, we must receive the conclusions drawn from them as the general laws of hydraulics.

The experiments of the abbe Bossut, narrated in his Hydrodynamique, are of the greatest value, having been made in the cases of most general frequency, and being made with great care. The greatest service, however, has been done by the chevalier Buat, who saw the folly of attempting to deduce an accurate theory from any principles that we have as yet learned, and the necessity of adhering to such a theory as could be deduced from experiment alone, independent of any more general principles. Such a theory must be a just one, if the experiments are really general, unaffected by the particular circumstances of the case, and if the classes of experiment are sufficiently comprehensive to include all the cases which occur in the most important practical questions. Some principle was necessary, however, for connecting these experiments. The sufficiency of this principle was not easily ascertained. Mr. Buat's way of establishing this was judicious. If the principle is ill-founded, the results of its combination in cases of actual experiments must be irregular; but if experiments, seemingly very unlike, and in a vast variety of dissimilar cases, give a train of results which is extremely regular and consistent, we may presume that the principle, which in this manner harmonizes and reconciles things so unlike, is founded in the nature of things; and if this principle be such as is agreeable to our clearest notions of the internal mechanism of the motions of fluids, our presumption approaches to conviction.

Proceeding in this way, the chevalier Buat has collected a prodigious number of facts, comprehending almost every case of the motion of fluids. He first classed them according to their resemblance in some one particular, and observed the differences which accompanied their differences in other circumstances; and by considering what could produce these differences, he obtained general rules, deduced from fact, by which these differences could be made to fall into a regular series. He then arranged all the experiments under some other circumstances of resemblance, and pursued the same method; and by following this out, he has produced a general proposition, which applies to the whole of this numerous list of experiments with a precision far exceeding our utmost hopes. This proposition is contained in No. 59. of the article Rivers, and is there offered as one of the most valuable results of modern science.

We must, however, observe, that of this list of experiments there is a very large class, which is not direct, but requires a good deal of reflection to enable us to draw a confident conclusion; and this is in cases which are very frequent and important, viz. where the declivity is exceedingly small, as in open canals and rivers. The experiments were of the following forms: Two large cisterns were made to communicate with each other by means of a pipe. The surfaces of the water in these cisterns were made to differ only by a small fraction of an inch: and it is supposed that the motion in the communicating pipe will be the same as in a very long pipe, or an open canal, having this very minute declivity. We have no difficulty in admitting the conclusion; but we have seen it contested, and it is by no means intuitive. We had entertained hopes that this important case would have been determined by direct experiment, which the writer of this article was commissioned to make by the Board for Encouraging Improvements and Manufactures in Scotland: But the unfavourable state of his health was always an effectual bar to the accomplishment of this desirable object. This, however, need not occasion any hesitation in the adoption of M. Buat's general proposition, because the experiments which we are now criticising fall in precisely with the general train of the rest, and show no general deviation which would indicate a fallacy in principle.

We apprehend it to be quite unnecessary to add much to what has been already delivered on the motion of waters in an open canal. Their general progressive motion, and consequently the quantity delivered by an aqueduct of any slope and dimension, are sufficiently determined; and all that is wanted is the tables which we promised in No. 65. of the article Rivers, by which any person who understands common arithmetic may, in five minutes' time or less, compute the quantity of water which will be delivered by the aqueduct, canal, conduit, or pipe; for the theorem in No. 59. of this article applies to them all without distinction. We therefore take this opportunity of inserting these tables, which have been computed on purpose for this work with great labour. ### Table I. Logarithms of the Values of the Numerator of the Fraction $\frac{307}{\sqrt{s-1}} \left(\frac{\sqrt{d}-0.1}{s+1.6}\right)$ for every Value of the Hydraulic mean Depth $d$: Also the Value of $0.3 \left(\frac{\sqrt{d}-0.1}{s+1.6}\right)$.

| $d$ | Log of $\frac{307}{\sqrt{d}-0.1}$ | $c_1 X$ | $d$ | Log of $\frac{307}{\sqrt{d}-0.1}$ | $c_2 X$ | $d$ | Log of $\frac{307}{\sqrt{d}-0.1}$ | $c_3 X$ | $d$ | Log of $\frac{307}{\sqrt{d}-0.1}$ | $c_4 X$ | |-----|----------------------------------|--------|-----|----------------------------------|--------|-----|----------------------------------|--------|-----|----------------------------------|--------| | 0.1 | 1.82208 | 0.06 | 4.9 | 2.81216 | 0.63 | 9.7 | 2.96634 | 0.9 | 54 | 3.34738 | 2.17 | | 0.2 | 2.02786 | 0.1 | 5.0 | 2.81674 | 0.63 | 9.8 | 2.96865 | 0.91 | 55 | 3.35143 | 2.19 | | 0.3 | 2.13753 | 0.13 | 5.1 | 2.82125 | 0.63 | 9.9 | 2.97093 | 0.91 | 56 | 3.35539 | 2.21 | | 0.4 | 2.21343 | 0.16 | 5.2 | 2.82507 | 0.65 | 10 | 2.97319 | 0.92 | 57 | 3.35928 | 2.23 | | 0.5 | 2.27049 | 0.18 | 5.3 | 2.83000 | 0.66 | 11 | 2.99434 | 0.97 | 58 | 3.36312 | 2.25 | | 0.6 | 2.31618 | 0.2 | 5.4 | 2.83222 | 0.67 | 12 | 3.01401 | 1.01 | 59 | 3.36687 | 2.27 | | 0.7 | 2.35441 | 0.22 | 5.5 | 2.83840 | 0.67 | 13 | 3.03189 | 1.05 | 60 | 3.37037 | 2.3 | | 0.8 | 2.38719 | 0.24 | 5.6 | 2.84248 | 0.68 | 14 | 3.04843 | 1.09 | 61 | 3.37421 | 2.31 | | 0.9 | 2.41588 | 0.25 | 5.7 | 2.84648 | 0.68 | 15 | 3.06383 | 1.13 | 62 | 3.37778 | 2.33 | | 1.0 | 2.44138 | 0.27 | 5.8 | 2.85043 | 0.69 | 16 | 3.07820 | 1.17 | 63 | 3.38130 | 2.35 | | 1.1 | 2.46431 | 0.28 | 5.9 | 2.85431 | 0.69 | 17 | 3.09170 | 1.21 | 64 | 3.38477 | 2.37 | | 1.2 | 2.48618 | 0.3 | 6.0 | 2.85812 | 0.7 | 18 | 3.10441 | 1.24 | 65 | 3.38817 | 2.39 | | 1.3 | 2.50426 | 0.31 | 6.1 | 2.86185 | 0.7 | 19 | 3.11644 | 1.28 | 66 | 3.39158 | 2.41 | | 1.4 | 2.52185 | 0.32 | 6.2 | 2.86554 | 0.71 | 20 | 3.12783 | 1.31 | 67 | 3.39483 | 2.42 | | 1.5 | 2.53818 | 0.34 | 6.3 | 2.86916 | 0.72 | 21 | 3.13867 | 1.34 | 68 | 3.39809 | 2.44 | | 1.6 | 2.55345 | 0.35 | 6.4 | 2.87271 | 0.73 | 22 | 3.14899 | 1.38 | 69 | 3.40130 | 2.46 | | 1.7 | 2.56769 | 0.36 | 6.5 | 2.87622 | 0.73 | 23 | 3.15885 | 1.41 | 70 | 3.40446 | 2.48 | | 1.8 | 2.58112 | 0.37 | 6.6 | 2.87966 | 0.74 | 24 | 3.16838 | 1.44 | 71 | 3.40758 | 2.49 | | 1.9 | 2.59381 | 0.38 | 6.7 | 2.88306 | 0.75 | 25 | 3.17734 | 1.47 | 72 | 3.41065 | 2.51 | | 2.0 | 2.60580 | 0.39 | 6.8 | 2.88641 | 0.75 | 26 | 3.18601 | 1.5 | 73 | 3.41369 | 2.53 | | 2.1 | 2.61713 | 0.4 | 6.9 | 2.88971 | 0.76 | 27 | 3.19438 | 1.53 | 74 | 3.41667 | 2.55 | | 2.2 | 2.62803 | 0.41 | 7.0 | 2.89296 | 0.76 | 28 | 3.20243 | 1.56 | 75 | 3.41962 | 2.57 | | 2.3 | 2.63839 | 0.42 | 7.1 | 2.89614 | 0.77 | 29 | 3.21020 | 1.58 | 76 | 3.42253 | 2.58 | | 2.4 | 2.64827 | 0.44 | 7.2 | 2.89930 | 0.77 | 30 | 3.21770 | 1.61 | 77 | 3.42540 | 2.6 | | 2.5 | 2.65772 | 0.45 | 7.3 | 2.90241 | 0.78 | 31 | 3.22495 | 1.64 | 78 | 3.42823 | 2.62 | | 2.6 | 2.66681 | 0.45 | 7.4 | 2.90549 | 0.78 | 32 | 3.23196 | 1.67 | 79 | 3.43103 | 2.63 | | 2.7 | 2.67536 | 0.46 | 7.5 | 2.90851 | 0.79 | 33 | 3.23877 | 1.69 | 80 | 3.43380 | 2.65 | | 2.8 | 2.68395 | 0.47 | 7.6 | 2.91150 | 0.79 | 34 | 3.24537 | 1.72 | 81 | 3.43653 | 2.67 | | 2.9 | 2.69207 | 0.48 | 7.7 | 2.91445 | 0.8 | 35 | 3.25176 | 1.74 | 82 | 3.43923 | 2.69 | | 3.0 | 2.69989 | 0.49 | 7.8 | 2.91734 | 0.8 | 36 | 3.25799 | 1.77 | 83 | 3.44189 | 2.7 | | 3.1 | 2.70743 | 0.5 | 7.9 | 2.92022 | 0.81 | 37 | 3.26404 | 1.79 | 84 | 3.44452 | 2.72 | | 3.2 | 2.71472 | 0.51 | 8.0 | 2.92305 | 0.82 | 38 | 3.26993 | 1.82 | 85 | 3.44712 | 2.74 | | 3.3 | 2.72181 | 0.52 | 8.1 | 2.92584 | 0.82 | 39 | 3.27566 | 1.84 | 86 | 3.44968 | 2.75 | | 3.4 | 2.72866 | 0.53 | 8.2 | 2.92860 | 0.83 | 40 | 3.28125 | 1.87 | 87 | 3.45222 | 2.77 | | 3.5 | 2.73531 | 0.53 | 8.3 | 2.93133 | 0.83 | 41 | 3.28669 | 1.89 | 88 | 3.45473 | 2.78 | | 3.6 | 2.74178 | 0.54 | 8.4 | 2.93403 | 0.84 | 42 | 3.29201 | 1.91 | 89 | 3.45721 | 2.79 | | 3.7 | 2.74805 | 0.55 | 8.5 | 2.93670 | 0.84 | 43 | 3.29720 | 1.93 | 90 | 3.45965 | 2.81 | | 3.8 | 2.75417 | 0.56 | 8.6 | 2.93933 | 0.85 | 44 | 3.30227 | 1.95 | 91 | 3.46208 | 2.83 | | 3.9 | 2.76009 | 0.56 | 8.7 | 2.94192 | 0.85 | 45 | 3.30722 | 1.98 | 92 | 3.46448 | 2.85 | | 4.0 | 2.76589 | 0.57 | 8.8 | 2.94449 | 0.86 | 46 | 3.31207 | 2.00 | 93 | 3.46685 | 2.86 | | 4.1 | 2.77153 | 0.58 | 8.9 | 2.94703 | 0.86 | 47 | 3.31681 | 2.03 | 94 | 3.46920 | 2.88 | | 4.2 | 2.77704 | 0.59 | 9.0 | 2.94954 | 0.87 | 48 | 3.32145 | 2.05 | 95 | 3.47152 | 2.89 | | 4.3 | 2.78240 | 0.59 | 9.1 | 2.95202 | 0.87 | 49 | 3.32599 | 2.07 | 96 | 3.47381 | 2.91 | | 4.4 | 2.78765 | 0.6 | 9.2 | 2.95447 | 0.88 | 50 | 3.33043 | 2.09 | 97 | 3.47608 | 2.93 | | 4.5 | 2.79277 | 0.6 | 9.3 | 2.95690 | 0.88 | 51 | 3.33485 | 2.11 | 98 | 3.47833 | 2.94 | | 4.6 | 2.79779 | 0.61 | 9.4 | 2.95930 | 0.89 | 52 | 3.33908 | 2.13 | 99 | 3.48056 | 2.95 | | 4.7 | 2.80269 | 0.62 | 9.5 | 2.96167 | 0.89 | 53 | 3.34327 | 2.15 | 100 | 3.48277 | 2.97 | | 4.8 | 2.80747 | 0.63 | 9.6 | 2.96402 | 0.9 | | | | | | | ### Table II. Logarithms of the Values of the Denominator of the Fraction $\frac{397(\sqrt{d-1.8})}{\sqrt{s}-L\sqrt{s+1.6}}$ for every Value of the Slopes.

| $s/\sqrt{1-L\sqrt{1-s}}$ | Log. of $s/\sqrt{1-L\sqrt{1-s}}$ | $s/\sqrt{1-L\sqrt{1-s}}$ | Log. of $s/\sqrt{1-L\sqrt{1-s}}$ | $s/\sqrt{1-L\sqrt{1-s}}$ | Log. of $s/\sqrt{1-L\sqrt{1-s}}$ | $s/\sqrt{1-L\sqrt{1-s}}$ | Log. of $s/\sqrt{1-L\sqrt{1-s}}$ | |--------------------------|----------------------------------|--------------------------|----------------------------------|--------------------------|----------------------------------|--------------------------|----------------------------------| | 1.0 | 9.7194 | 7.3 | 0.2651 | 48 | 0.67997 | 1.70 | 1.01983 | | 1.2 | 9.7421 | 7.4 | 0.2997 | 46 | 0.68574 | 1.80 | 1.03410 | | 1.3 | 9.7836 | 7.6 | 0.2136 | 48 | 0.69135 | 1.90 | 1.04751 | | 1.4 | 9.8022 | 7.7 | 0.2109 | 49 | 0.70226 | 2.00 | 1.06026 | | 1.5 | 9.8188 | 7.8 | 0.2233 | 50 | 0.70749 | 2.10 | 1.08390 | | 1.6 | 9.8341 | 7.9 | 0.2263 | 51 | 0.71263 | 2.20 | 1.09489 | | 1.7 | 9.8493 | 8.0 | 0.2298 | 52 | 0.71767 | 2.30 | 1.10542 | | 1.8 | 9.8631 | 8.1 | 0.2329 | 53 | 0.72263 | 2.40 | 1.11553 | | 1.9 | 9.8762 | 8.2 | 0.2361 | 54 | 0.72746 | 2.50 | 1.12523 | | 2.0 | 9.8885 | 8.3 | 0.2392 | 55 | 0.73223 | 2.60 | 1.13453 | | 2.1 | 9.9003 | 8.4 | 0.2422 | 56 | 0.73695 | 2.70 | 1.14345 | | 2.2 | 9.9113 | 8.5 | 0.2452 | 57 | 0.74155 | 2.80 | 1.15204 | | 2.3 | 9.9226 | 8.6 | 0.2482 | 58 | 0.74601 | 2.90 | 1.16235 | | 2.4 | 9.9327 | 8.7 | 0.2519 | 59 | 0.75043 | 3.00 | 1.17632 | | 2.5 | 9.9423 | 8.8 | 0.2543 | 60 | 0.75481 | 3.10 | 1.17612 | | 2.6 | 9.9517 | 8.9 | 0.2570 | 61 | 0.75906 | 3.20 | 1.18363 | | 2.7 | 9.9608 | 9.0 | 0.2596 | 62 | 0.76328 | 3.30 | 1.19292 | | 2.8 | 9.9694 | 9.1 | 0.2621 | 63 | 0.76745 | 3.40 | 1.19803 | | 2.9 | 9.9781 | 9.2 | 0.2660 | 64 | 0.77151 | 3.50 | 1.20492 | | 3.0 | 9.9863 | 9.3 | 0.2689 | 65 | 0.77576 | 3.60 | 1.21148 | | 3.1 | 9.9942 | 9.4 | 0.2711 | 66 | 0.77945 | 3.70 | 1.21806 | | 3.2 | 0.0020 | 9.5 | 0.2738 | 67 | 0.78333 | 3.80 | 1.22435 | | 3.3 | 0.0045 | 9.6 | 0.2766 | 68 | 0.78718 | 3.90 | 1.23048 | | 3.4 | 0.0166 | 9.7 | 0.2792 | 69 | 0.79092 | 4.00 | 1.23647 | | 3.5 | 0.0273 | 9.8 | 0.2816 | 70 | 0.79463 | 4.10 | 1.24232 | | 3.6 | 0.0304 | 9.9 | 0.2846 | 71 | 0.79824 | 4.20 | 1.24835 | | 3.7 | 0.0373 | 10 | 0.2879 | 72 | 0.80182 | 4.30 | 1.25362 | | 3.8 | 0.0438 | 11 | 0.2910 | 73 | 0.80536 | 4.40 | 1.25923 | | 3.9 | 0.0515 | 12 | 0.2942 | 74 | 0.80882 | 4.50 | 1.26433 | | 4.0 | 0.0588 | 13 | 0.2976 | 75 | 0.81231 | 4.60 | 1.26951 | | 4.1 | 0.0624 | 14 | 0.3008 | 76 | 0.81571 | 4.70 | 1.27461 | | 4.2 | 0.0689 | 15 | 0.3040 | 77 | 0.81908 | 4.80 | 1.27957 | | 4.3 | 0.0741 | 16 | 0.3073 | 78 | 0.82236 | 4.90 | 1.28445 | | 4.4 | 0.0789 | 17 | 0.3105 | 79 | 0.82562 | 5.00 | 1.28903 | | 4.5 | 0.0833 | 18 | 0.3137 | 80 | 0.82885 | 5.10 | 1.29391 | | 4.6 | 0.0982 | 19 | 0.3170 | 81 | 0.83226 | 5.20 | 1.29851 | | 4.7 | 0.0965 | 20 | 0.3202 | 82 | 0.83525 | 5.30 | 1.30300 | | 4.8 | 0.1058 | 21 | 0.3245 | 83 | 0.83835 | 5.40 | 1.30740 | | 4.9 | 0.1074 | 22 | 0.3279 | 84 | 0.84142 | 5.50 | 1.31172 | | 5.0 | 0.1114 | 23 | 0.3312 | 85 | 0.84442 | 5.60 | 1.31597 | | 5.1 | 0.1163 | 24 | 0.3345 | 86 | 0.84739 | 5.70 | 1.32015 | | 5.2 | 0.1208 | 25 | 0.3378 | 87 | 0.85034 | 5.80 | 1.32426 | | 5.3 | 0.1259 | 26 | 0.3411 | 88 | 0.85327 | 5.90 | 1.32832 | | 5.4 | 0.1306 | 27 | 0.3445 | 89 | 0.85618 | 6.00 | 1.33226 | | 5.5 | 0.1351 | 28 | 0.3478 | 90 | 0.85908 | 6.10 | 1.33594 | | 5.6 | 0.1397 | 29 | 0.3512 | 91 | 0.86189 | 6.20 | 1.33997 | | 5.7 | 0.1440 | 30 | 0.3546 | 92 | 0.86463 | 6.30 | 1.34373 | | 5.8 | 0.1484 | 31 | 0.3579 | 93 | 0.86741 | 6.40 | 1.34743 | | 5.9 | 0.1527 | 32 | 0.3612 | 94 | 0.87017 | 6.50 | 1.35108 | | 6.0 | 0.1569 | 33 | 0.3645 | 95 | 0.87286 | 6.60 | 1.35468 | | 6.1 | 0.1613 | 34 | 0.3679 | 96 | 0.87552 | 6.70 | 1.35823 | | 6.2 | 0.1652 | 35 | 0.3712 | 97 | 0.87818 | 6.80 | 1.36170 | | 6.3 | 0.1692 | 36 | 0.3745 | 98 | 0.88076 | 6.90 | 1.36513 | | 6.4 | 0.1732 | 37 | 0.3779 | 99 | 0.88338 | 7.00 | 1.36851 | | 6.5 | 0.1771 | 38 | 0.3812 | 100 | 0.88593 | 7.10 | 1.37185 | | 6.6 | 0.1809 | 39 | 0.3846 | 110 | 0.89104 | 7.20 | 1.37513 | | 6.7 | 0.1847 | 40 | 0.3879 | 120 | 0.90141 | 7.30 | 1.37839 | | 6.8 | 0.1885 | 41 | 0.3912 | 130 | 0.91212 | 7.40 | 1.38157 | | 6.9 | 0.1922 | 42 | 0.3946 | 140 | 0.91709 | 7.50 | 1.38471 | | 7.0 | 0.1958 | 43 | 0.3983 | 150 | 0.92843 | 7.60 | 1.39082 | | 7.1 | 0.1983 | 44 | 0.4016 | 160 | 0.94066 | 7.70 | 1.39301 | | 7.2 | 0.1988 | 160 | 1.00166 | 790 | TABLE I. consists of three columns.—Column 1, entitled \(d\), contains the hydraulic mean depths of any conduit in inches. This is set down for every tenth of an inch in the first 10 inches, that the answers may be more accurately obtained for pipes, the mean depth of which seldom exceeds three or four inches. The column is continued to 100 inches, which is fully equal to the hydraulic mean depth of any canal.

Column 2. contains the logarithms of the values of \(\sqrt{d} = 0.1\), multiplied by 307, that is, the logarithm of the numerator of the fraction \(\frac{307}{\sqrt{s - 1}}\) in No. 65. of the article RIVERS.

Column 3. contains the product of the values of \(\sqrt{d} = 0.1\) multiplied by 0.3.

TABLE II. consists of two columns.—Column 1, entitled \(s\), contains the denominator of the fraction expressing the slope or declivity of any pipe or canal; that is, the quotient of its length divided by the elevation of one extremity above the other. Thus, if a canal of one mile in length be three feet higher at one end than the other, then \(s = \frac{5280}{3} = 1760\).

Column 2. contains the logarithms of the denominators of the above-mentioned fraction, or of the different values of the quantity \(\sqrt{s - 1}\).

These quantities were computed true to the third decimal place. Notwithstanding this, the last figure in about a dozen of the first logarithms of each table is not absolutely certain to the nearest unit. But this cannot produce an error of 1 in 100,000.

Examples of the Use of these Tables.

Example 1. Water is brought into the city of Edinburgh in several mains. One of these is a pipe of five inches diameter. The length of the pipe is 14,637 feet; and the reservoir at Comiston is 44 feet higher than the reservoir into which it delivers the water on the Castle Hill. Query, The number of Scotch pints which this pipe should deliver in a minute?

1. We have \(d = \frac{5}{4} = 1.25\) inches. The logarithm corresponding to this \(d\), being nearly the mean between the logarithms corresponding to 1.2 and 1.3 is 2.49472.

2. We have \(s = \frac{14637}{44} = 332.7\). The logarithm corresponding to this in Table II, is had by taking proportional parts for the difference between the logarithms for \(s = 330\) and \(s = 340\), and is 1.18533.

3. From 2.49472 Take 1.18533

Remains 1.30939, the logarithm of 20.385 inches.

4. In column 3. of Table I, opposite to \(d = 1.2\) and \(d = 1.3\) are 0.3 and 0.31, of which the mean is 0.305 inches, the correction for viscosity.

5. Therefore the velocity in inches per second is \(20.385 - 0.305 = 20.08\).

6. To obtain the Scotch pints per minute (each containing 103.4 cubic inches), multiply the velocity by 60, and this product by 5, and this by 0.7854 (the area of a circle whose diameter is 1), and divide by 103.4. Or, by logarithms,

Add the log. of 20.08 - 1.30276 log. of 60 - 1.77815 log. of 5 or 25 - 1.39794 log. of 0.7854 - 9.80509

Subtract the log. of 103.4 - 4.37394

Remains the log. of 228.8 pints - 2.35943

Example 2. The canal mentioned in the article RIVERS, No. 63, was 18 feet broad at the surface, and 7 feet at the bottom. It was 4 feet deep, and had a declivity of 4 inches in a mile. Query, The mean velocity?

1. The slant side of the canal, corresponding to 4 feet deep and 5½ projection, is 6.8 feet; therefore the border touched by the water is 6.8 + 7 + 6.8 = 20.6. The area is \(4 \times \frac{18 + 7}{2} = 50\) square feet. Therefore \(d = \frac{50}{20.6} = 2.427\) feet, or 29.124 inches. The logarithm corresponding to this in Table I. is 3.21113, and the correction for viscosity from the third column of the same Table is 1.58.

2. The slope is one-third of a foot in a mile, or one foot in three miles. Therefore \(s = 15.840\). The logarithm corresponding to this is 2.08280.

3. From 3.21113 Subtract 2.08280

Remains 1.12833 = log. of 13.438 inches.

Subtract for viscosity 1.58

Velocity per second - 11.858

This velocity is considerably smaller than what was observed by Mr Watt. And indeed we observe, that in the very small declivities of rivers and canals, the formula is a little different. We have made several comparisons with a formula which is essentially the same with Buat's, and comes nearer in these cases. Instead of taking the hyperbolic logarithm of \(\sqrt{s + 1.6}\), multiply its common logarithm by 24, or multiply it by 9, and divide the product by 4; and this process is vastly easier than taking the hyperbolic logarithm.

We have not, however, presumed to calculate tables on the authority of our own observations, thinking too respectfully of this gentleman's labours and observations. But this subject will, ere long, be fully established on a series of observations on canals of various dimensions and declivities, made by several eminent engineers during the execution of them. Fortunately Mr Buat's formula is chiefly founded on observations on small canals; and is therefore most accurate in such works where it is most necessary, viz. in mill courses, and other derivations for working machinery.

We now proceed to take notice of a few circumstances which deserve attention, in the construction of canals, in addition to those delivered in the article RIVERS.

When a canal or aqueduct is brought off from a basin or larger stream, it ought always to be widened at the entry, if it is intended for drawing off a continued stream of water: For such a canal has a slope, without which it can have no current. Suppose it filled to a dead level to the farther end: Take away the bar, and the water immediately begins to flow off at that end. But it is some time before any motion is perceived at the head of the canal, during all which time the motion of the water is augmenting in every part of the canal; consequently the slope is increasing in every part, this being the sole cause of its stream. When the water at the entry begins to move, the slope is scarcely sensible there; but it sensibly steepens every moment with the increase of velocity, which at last attains its maximum relative to the slope and dimensions of the whole canal; and this regulates the depth of water in every point down the stream. When all has attained a state of permanency, the slope at the entry remains much greater than in any other part of the canal; for this slope must be such as will produce a velocity sufficient for supplying its train.

And it must be remembered, that the velocity which must be produced greatly exceeds the mean velocity corresponding to the train of the canal. Suppose that this is 25 inches. There must be a velocity of 30 inches at the surface, as appears by the Table in the article Rivers, No. 8o. This must be produced by a real fall at the entry.

In every other part the slope is sufficient, if it merely serves to give the water (already in motion) force enough for overcoming the friction and other resistances. But at the entry the water is stagnant, if in a basin, or it is moving past laterally, if the aqueduct is derived from a river; and, having no velocity whatever in the direction of the canal, it must derive it from its slope. The water therefore which has acquired a permanent form in such an aqueduct, must necessarily take that form which exactly performs the offices requisite in its different portions. The surface remains horizontal in the basin, as to K.C (fig. 1.), till it comes near the entry of the canal A.B., and there it acquires the form of an undulated curve C.D.F.; and then the surface acquires an uniform slope E.F., in the lower part of the canal, where the water is in train.

If this is a drain, the discharge is much less than might be produced by the same bed if this sudden slope could be avoided. If it is to be navigated, having only a very gentle slope in its whole length, this sudden slope is a very great imperfection, both by diminishing the depth of water, which might otherwise be obtained along the canal, and by rendering the passage of boats into the basin very difficult, and the coming out very hazardous.

All this may be avoided, and the velocity at the entry may be kept equal to that which forms the train of the canal, by the simple process of enlarging the entry. Suppose that the water could accelerate along the slopes of the canal, as a heavy body would do on a finely polished plane. If we now make the width of the entry in its different parts inversely proportional to the fictitious velocities in those parts, it is plain that the slope of the surface will be made parallel to that of the canal which is in train. This will require a form somewhat like a bell or speaking trumpet, as may easily be shown by a mathematical discussion. It would, however, be so much evasated at the basin as to occupy much room, and it would be very expensive to make such an excavation. But we may, at a very moderate expense of money and room, make the increase of velocity at the entry almost insensible. This should always be done, and it is not all expense: for if it be not done, the water will undermine the banks on each side, because it is moving very swiftly, and will make an excavation for itself, leaving all the mud in the canal below. We may observe this enlargement at the entry of all natural derivations from a basin or lake. It is a very instructive experiment, to fill up this enlargement, continuing the parallel sides of the drain quite to the side of the lake. We shall immediately observe the water grow shallower in the drain, and its performance will diminish. Supposing the ditch carried on with parallel sides quite to the side of the basin, if we build two walls or dykes from the extremities of those sides, bending outwards with a proper curvature (and this will often be less costly than widening the drain), the discharge will be greatly increased. We have seen instances where it was nearly doubled.

The enlargement at the mouths of rivers is generally owing to the same cause. The tide of flood up the river produces a superficial slope opposite to that of the river, and this widens the mouth. This is most remarkable when the tides are high, and the river has little slope.

After this great fall at the entry of the canal, in which all the filaments are much accelerated, and the inferior ones most of all, things take a contrary turn. The water, by rubbing on the bottom and the sides, is retarded; and therefore the section must, from being shallow, become a little deeper, and the surface will be convex for some distance till all comes into train. When this is established, the filaments nearest the bottom and side are moving slowest, and the surface (in the middle especially) retains the greatest velocity, gliding over the rest. The velocity in the canal, and the depth of the section, adjust themselves in such a manner that the difference between the surface of the basin and the surface of the uniform section of the canal corresponds exactly to the velocity. Thus, if this be observed to be two feet in a second, the difference of height will be \( \frac{1}{16} \) of an inch.

All the practical questions that are of considerable importance respecting the motion of water in aqueducts, may be easily, though not elegantly, solved by means of the tables.

But it is to be remembered, that these tables relate only to uniform motion, that is, to water that is in train, and where the velocity suffers no change by lengthening the conduit, provided the slope remain the same. It is much more difficult to determine what will be the velocity, &c. in a canal of which nothing is given but the form, and slope, and depth of the entry, without saying how deep the water runs in it. And it is here that the common doctrines of hydraulics are most in fault, and unable to teach us how deep the water will run in a canal, though the depth of the basin at the entry be perfectly known. Between the part of the canal which is in train and the basin, there is an interval where the water is in a state of acceleration, and is afterwards retarded.

The determination of the motions in this interval is exceedingly exceedingly difficult, even in a rectangular canal. It was one great aim of M. Boult's experiments to ascertain this by measuring accurately the depth of the water. But he found that when the slope was but a very few inches in the whole length of his canal, it was not in train for want of greater length; and when the slope was still less, the small fractions of an inch, by which he was to judge of the variations of depth, could not be measured with sufficient accuracy. It would be a most desirable point to determine the length of a canal, whose slope and other dimensions are given, which will bring it into train; and what is the ratio which will then obtain between the depth at the entry and the depth which will be maintained. Till this be done, the engineer cannot ascertain by a direct process what quantity of water will be drawn off from a reservoir by a given canal. But as yet this is out of our reach. Experiments, however, are in view which will promote the investigation.

But this and similar questions are of such importance, that we cannot be said to have improved hydraulics, unless we can give a tolerably precise answer. This we can do by a sort of retrograde process, proceeding on the principles of uniform motion established by the Chevalier Boult. We may suppose a train maintained in the canal, and then examine whether this train can be produced by any fall that is possible at the entry. If it can, we may be certain that it is so produced, and our problem is solved.

We shall now point out the methods of answering some chief questions of this kind.

**Quest. 1.** Given the slope $s$ and the breadth $w$ of a canal, and the height $H$ of the surface of the water in the basin above the bottom of the entry; to find the depth $h$ and velocity $V$ of the stream, and the quantity of water $Q$ which is discharged?

The chief difficulty is to find the depth of the stream where it is in train. For this end, we may simplify the hydraulic theorem of uniform motion in No. 59. of the article River; making $V = \frac{\sqrt{N_g d}}{\sqrt{s}}$, where $g$ is the velocity (in inches) acquired in a second by falling, $d$ is the hydraulic mean depth, and $\sqrt{s}$ stands for $\sqrt{s} - L \cdot \sqrt{s + 1.6}$. $N$ is a number to be fixed by experiment (see River, No. 53.) depending on the contraction or obstruction sustained at the entry of the canal, and it may in most common cases be taken = 244; so that $\sqrt{N_g}$ may be somewhat less than 307. To find it, we may begin by taking for our depth of stream a quantity $h$, somewhat smaller than $H$ the height of the surface of the basin above the bottom of the canal. With this depth, and the known width $w$ of the canal, we can find the hydraulic depth $d$ (See Rivers, No. 48). Then with $\sqrt{d}$ and the slope find $V$ by the Table: make this $V = \frac{\sqrt{N_g d}}{\sqrt{s}}$. This gives $\sqrt{N_g} = \frac{V}{\sqrt{s}}$.

This value of $N_g$ is sufficiently exact; for a small error of depth hardly affects the hydraulic mean depth.

After this preparation, the expression of the mean velocity in the canal will be $\frac{\sqrt{N_g w h}}{\sqrt{s}}$. The height which will produce this velocity is $\frac{N_g}{2G} \left( \frac{w h}{w + 2h} \right)$.

Now this is the slope at the entry of the canal which produces the velocity that is afterwards maintained against the obstructions by the slope of the canal. It is therefore $= H - h$. Hence we deduce

$$h = \frac{-\left( \frac{w}{2G} \left( \frac{N_g}{2G} + 1 \right) - 2H \right)}{4}$$

$$+ \frac{\sqrt{8Hw + \left( \frac{w}{2G} \left( \frac{N_g}{2G} + 1 \right) - 2H \right)^2}}{4}$$

If there be no contraction at the entry, $g = G$ and $\frac{g}{2G} = \frac{1}{2}$.

Having thus obtained the depth $h$ of the stream, we obtain the quantity of water by combining this with the width $w$ and the velocity $V$.

But as this was but an approximation, it is necessary to examine whether the velocity $V$ be possible. This is very easy. It must be produced by the fall $H - h$. We shall have no occasion for any correction of our first assumption, if $h$ has not been extravagantly erroneous, because a small mistake in $h$ produces almost the same variation in $d$. The test of accuracy, however, is, that $h$, together with the height which will produce the velocity $V$, must make up the whole height $H$. Assuming $h$ too small, leaves $H - h$ too great, and will give a small velocity $V$, which requires a small value of $H - h$. The error of $H - h$ therefore is always greater than the error we have committed in our first assumption. Therefore when this error of $H - h$ is but a trifle, such as one-fourth of an inch, we may rest satisfied with our answer.

Perhaps the easiest process may be the following: Suppose the whole stream in train to have the depth $H$. The velocity $V$ obtained for this depth and slope by the Table requires a certain productive height $u$. Make $\sqrt{H + u} : H = H : h$, and $h$ will be exceedingly near the truth. The reason is obvious.

**Quest. 2.** Given the discharge (or quantity to be furnished in a second) $Q$, the height $H$ of the basin above the bottom of the canal, and the slope; to find the dimensions of the canal?

Let $x$ and $y$ be the depth and mean width. It is plain that the equation $\frac{Q}{xy} = \frac{\sqrt{2G}}{\sqrt{H - x}}$ will give a value of $y$ in terms of $x$. Compare this with the value of $y$ obtained from the equation $\frac{Q}{xy} = \frac{\sqrt{N_g}}{\sqrt{s}} \cdot \frac{x}{y + 2x}$. This will give an equation containing only $x$ and known quantities. But it will be very complicated, and we must have recourse to an approximation. This will be best understood in the form of an example.

Suppose the depth at the entry to be 18 inches, and the slope $\frac{1}{1000}$. Let 1200 cubic feet of water per minute be the quantity of water to be drawn off, for working machinery or any other purpose; and let the canal... be supposed of the best form, recommended in No. 69. of the article RIVER, where the base of the sloping side is four-thirds of the height.

The slightest consideration will show us that if \( \frac{V^4}{744} \) be taken for the height producing the velocity, it cannot exceed 3 inches, nor be less than 1. Suppose it =2, and therefore the depth of the stream in the canal to be 16 inches; find the mean width of the canal by the equation \( w = \frac{Q}{h(\sqrt{d-0.1}(\frac{327}{\sqrt{8}}-0.3)} \), in which Q is 20 cubic feet (the 60th part of 1200), \( \sqrt{8} = 28.153, \sqrt{1000} = 31.6, \) and \( h = 16 \). This gives \( w = 5.52 \) feet. The section \( n = 7.36 \) feet, and \( V = 32.6 \) inches. This requires a fall of 1.52 inches instead of 2 inches. Take this from 18, and there remains 16.28, which we shall find not to differ one-tenth of an inch from the exact depth which the water will acquire and maintain. We may therefore be satisfied with assuming 5.36 feet as the mean width, and 3.53 feet for the width at the bottom.

This approximation proceeds on this consideration, that when the width diminishes by a small quantity, and in the same proportion that the depth increases, the hydraulic mean depth remains the same, and therefore the velocity also remains, and the quantity discharged changes in the exact proportion of the section. Any minute error which may result from this supposition, may be corrected by increasing the fall producing the velocity, in the proportion of the first hydraulic mean depth to the mean depth corresponding to the new dimensions found for the canal. It will now become 1.53, and \( V \) will be 32.72, and the depth will be 16.47. The quantity discharged being divided by \( V \), will give the section \( = 7.335 \) feet, from which, and the new depth, we obtain 5.344 for the width.

This and the foregoing are the most common questions proposed to an engineer. We asserted with some confidence that few of the profession are able to answer them with tolerable precision. We cannot offend the professional gentlemen by this, when we inform them that the Academy of Sciences at Paris were occupied during several months with an examination of a plan proposed by M. Parcieux, for bringing the waters of the Yvette into Paris; and after the most mature consideration, gave in a report of the quantity of water which M. De Parcieux's aqueduct would yield, and that their report has been found erroneous in the proportion of at least 2 to 5: For the waters have been brought in, and exceed the report in this proportion. Indeed long after the giving in the report, M. Perrochet, the most celebrated engineer in France, affirmed that the dimensions proposed were much greater than were necessary, and said that an aqueduct of 5½ feet wide, and 3½ deep with a slope of 15 inches in a thousand fathoms, would have a velocity of 12 or 13 inches per second, which would bring in all the water furnished by the proposed sources. The great diminution of expense occasioned by the alteration encouraged the community to undertake the work. It was accordingly begun, and a part executed. The water was found to run with a velocity of near 19 inches when it was 3½ feet deep. M. Perrochet founded his computation on his own experience alone, acknowledging that he had no theory to instruct him. The work was carried no farther, it being found that the city could be supplied at a much smaller expense by steam engines erected by Boulton and Watt. But the facts which occurred in the partial execution of the aqueduct are very valuable. If M. Perrochet's aqueduct be examined by our general formula, \( s \) will be found \( = \frac{1}{100} \), and \( d = 18.72 \), from which we deduce the velocity \( = 18.7 \), agreeing with the observation with astonishing precision.

The experiments at Turin by Michelotti on canals were very numerous, but complicated with many circumstances which would render the discussion too long for this place. When cleared of these circumstances, which we have done with scrupulous care, they are also abundantly conformable to our theory of the uniform motion of running waters. But to return to our subject:

Should it be required to bring off at once from the basin a mill-course, having a determined velocity for driving an undershot wheel, the problem becomes easier, because the velocity and slope combined determine the hydraulic mean depth at once; and the depth of the stream will be had by means of the height which must be taken for the whole depth at the entry, in order to produce the required velocity.

In like manner, having given the quantity to be discharged, and the velocity and the depth at the entry, we can find the other dimensions of the channel; and the mean depth being found, we can determine the slope.

When the slope of a canal is very small, so that the depth of the uniform stream differs but a little from that at the entry, the quantity discharged is but small. But a great velocity, requiring a great fall at the entry, produces a great diminution of depth, and therefore it may not compensate for this diminution, and the quantity discharged may be smaller. Improbable as this may appear, it is not demonstrably false; and hence we may see the propriety of the following

**Question 3.** Given the depth \( H \) at the entry of a rectangular canal, and also its width \( w \); required the slope, depth, and velocity which will produce the greatest possible discharge?

Let \( x \) be the unknown depth of the stream. \( H-x \) is the productive fall, and the velocity is \( \sqrt{2G(H-x)} \). This multiplied by \( wx \) will give the quantity discharged. Therefore \( wx\sqrt{2G(H-x)} \) must be made a maximum. The common process for this will give the equation, \( 2H = 3x \), or \( x = \frac{2}{3}H \). The mean velocity will be \( \sqrt{2G(H-x)} \); the section will be \( \frac{2}{3}wH \), and the discharge \( = \frac{2}{3}\sqrt{2GwH\sqrt{\frac{2}{3}H}} \), and \( d = \frac{2}{3}wH \). With these data the slope is easily had by the formula for uniform motion.

If the canal is of the trapezoidal form, the investigation is more troublesome, and requires the resolution of a cubic equation.

It may appear strange that increasing the slope of a canal beyond the quantity determined by this problem can diminish the quantity of water conveyed. But one of these two things must happen; either the motion will not acquire uniformity in such a canal for want of length, length, or the discharge must diminish. Supposing, however, that it could augment, we can judge how far this can go. Let us take the extreme case by making the canal vertical. In this case it becomes a simple weir or wasteboard. Now the discharge of a wasteboard is $\frac{1}{4} \sqrt{2Gw(h^2 - (\frac{1}{4}h)^2)}$. The maximum determined by the preceding problem is to that of the wasteboard of the same dimensions as $H_{\frac{1}{4}} H; H_{\frac{1}{4}} - (\frac{1}{4} H)^{\frac{3}{2}},$ or as $H_{\frac{1}{4}} H; H_{\frac{1}{4}} H_{\frac{1}{4}} = 5773:646$, nearly $= 9:10$.

Having given the dimensions and slope of a canal, we can discover the relation between its expenditure and the time; or we can tell how much it will sink the surface of a pond in 24 hours, and the gradual progress of this effect; and this might be made the subject of a particular problem. But it is complicated and difficult. In cases where this is an interesting object, we may solve the question with sufficient accuracy, by calculating the expenditure at the beginning, supposing the basin kept full. Then from the known area of the pond, we can tell in what time this expenditure will sink an inch; do the same on the supposition that the water is one-third lower, and that it is two-thirds lower (noticing the contraction of the surface of the pond occasioned by this abstraction of its waters). Thus we shall obtain three rates of diminution, from which we can easily deduce the desired relation between the expenditure and the time.

Aqueducts derived from a basin or river are commonly furnished with a sluice at the entry. This changes exceedingly the state of things. The slope of the canal may be precisely such as will maintain the mean velocity of the water which passes under the sluice; in which case the depth of the stream is equal to that of the sluice, and the velocity is produced at once by the head of water above it. But if the slope is less than this, the velocity of the issuing water is diminished, and the water must rise in the canal. This must check the efflux at the sluice, and the water will be as it were stagnant above what comes through below it. It is extremely difficult to determine at what precise slope the water will begin to check the efflux. The contraction at the lower edge of the board hinders the water from attaining at once the whole depth which it acquires afterwards, when its velocity diminishes by the obstructions. While the regorging which these obstructions occasion does not reach back to the sluice, the efflux is not affected by it.—Even when it does reach to the sluice, there will be a less depth immediately behind it than farther down the canal, where it is in train; because the swift moving water which is next the bottom drags with it the regorged water which lies on it: but the canal must be rapid to make this difference of depth sensible. In ordinary canals, with moderate slopes and velocities, the velocity at the sluice may be safely taken as if it were that which corresponds to the difference of depths above and below the sluice, where both were in train.

Let therefore $H$ be the depth above the sluice, and $h$ the depth in the canal. Let $e$ be the elevation of the sluice above the sole, and let $b$ be its breadth. The discharge will be $eb\sqrt{H-h}\sqrt{2G}$ for the sluice, and

$$w\frac{N_g}{\sqrt{s}}\frac{wh}{w+2h}$$ for the canal. These must be the same. This gives the equation $eb\sqrt{H-h}\sqrt{2G} = w\frac{N_g}{\sqrt{s}}\frac{wh}{w+2h}$ containing the solution of all the questions which can be proposed. The only uncertainty is in the quantity $G$, which expresses the velocity competent to the passage of the water through the orifice, circumstanced as it is, namely, subjected to contraction. This may be regulated by a proper form given to the entry into this orifice. The contraction may be almost annihilated by making the masonry of a cycloidal form on both sides, and also at the lower edge of the sluice-board, so as to give the orifice a form resembling fig. 5, D, in the article Rivers. If the sluice is thin in the face of a basin, the contraction will reduce $2G$ to $296$. If the sluice be as wide as the canal, $2G$ will be nearly $500$.

**Question 4.** Given the head of water in the basin $H$, the breadth $b$, and the elevation $e$ of the sluice, and the breadth $w$ and slope $s$ of the canal, to find the depth $h$ of the stream, the velocity, and the discharge?

We must (as in Question 2) make a first supposition for $h$, in order to find the proper value of $d$. Then the equation $eb\sqrt{H-h}\sqrt{2G} = w\frac{N_g}{\sqrt{s}}$ gives $h$

$$\frac{Gc^2b^2}{w^2N_g} + \frac{\sqrt{Gc^2b^2H}}{w^2N_g} + \frac{(Gc^2b^2s)}{w^2N_gd}.$$ If this value shall differ considerably from the one which we assumed in order to begin the computation, make use of it for obtaining a new value of $d$, and repeat the operation. We shall rarely be obliged to perform a third operation.

The following is of frequent use:

**Question 5.** Given the dimensions and the slope, with the velocity and discharge of a river in its ordinary state, required the area or section of the sluice which will raise the waters to a certain height, still allowing the same quantity of water to pass through? Such an operation may render the rivers navigable for small craft or rafts above the sluice.

The problem is reduced to the determination of the size of orifice which will discharge this water with a velocity competent to the height to which the river is to be raised; only we must take into consideration the velocity of the water above the sluice, considering it as produced by a fall which makes a part of the height productive of the whole velocity at the sluice. Therefore $H$, in our investigation, must consist of the height to which we mean to raise the waters, and the height which will produce the velocity with which the waters approach the sluice: $h$, or the depth of the stream, is the ordinary depth of the river. Then (using the former symbols) we have $eb\sqrt{2G}\frac{wh}{\sqrt{2G}(H-h)} = Q$.

If the area of the sluice is known, and we would learn the height to which it will raise the river, we have $H-h = \frac{Q^2}{2Gc^2b^2}$ for the expression of the rise of the water. water above its ordinary level. But from this we must take the height which would produce the velocity of the river; so that if the sluice were as wide as the river, and were raised to the ordinary surface of the water,

\[ Q^2 = \frac{2G}{\rho} \]

which expresses the height that produces the velocity under the sluice, must be equal to the depth of the river, and \( H - h \) will be \( = 0 \).

The performance of aqueduct drains is a very important thing, and merits our attention in this place. While the art of managing waters, and of conducting them so as to answer our demands, renders us very important service by embellishing our habitations, or promoting our commercial intercourse, the art of draining creates as it were new riches, fertilizing tracts of bog or marsh, which was not only useless, but hurtful by its unwholesome exhalations, and converting them into rich pastures and gay meadows. A wild country, occupied by marshes which are inaccessible to herds or flocks, and serve only for the haunts of water-fowls, or the retreat of a few poor fishermen, when once it is freed from the waters in which it is drowned, opens its lap to receive the most precious seeds, is soon clothed in the richest garb, gives life and abundance to numerous herds, and never fails to become the delight of the industrious cultivator who has enfranchised it, and is attached to it by the labour which it cost him. In return, it procures him abundance, and supplies him with the means of daily augmenting its fertility. No species of agriculture exhibits such long-continued and progressive improvement. New families flock to the spot, and there multiply; and there nature seems the more eager to repay their labours, in proportion as she has been obliged, against her will, to keep her treasures locked up for a longer time, chilled by the waters. The countries newly inhabited by the human race, as is a great part of America, especially to the southward, are still covered to a great extent with marshes and lakes; and they would long remain in this condition, if population, daily making new advances, did not increase industry, by multiplying the cultivating hands, at the same time that it increases their wants. The Author of this beautiful world has at the beginning formed the great masses of mountain, has scooped out the dales and sloping hills, has traced out the courses, and even formed the beds of the rivers: but he has left to man the care of making his place of abode, and the field which must feed him, dry and comfortable. For this task is not beyond his powers, as the others are. Nay, by having this given to him in charge, he is richly repaid for his labour by the very state in which he finds those countries into which he penetrates for the first time. Being covered with lakes and forests, the juices of the soil are kept for him as it were in reserve. The air, the burning heat of the sun, and the continual washing of rains, would have combined to expend and dissipate their vegetative powers, had the fields been exposed in the same degree to their action as the inhabited and cultivated countries, the most fertile moulds of which are long since lodged in the bottom of the ocean. All this would have been completely lost through the whole extent of South America, had it not been protected by the forests which man must cut down, by the rank herbage which he must burn, and by the marsh and bog which he must destroy by draining. Let not ungrateful man complain of this. It is his duty to take on himself the task of opening up treasures, preserved on purpose for him with so much judgment and care. If he has discernment and sensibility, he will even thank the Author of all good, who has thus husbanded them for his use. He will co-operate with his beneficent views, and will be careful not to proceed by wantonly snatching at present any partial good, and by picking out what is most easily got at, regardless of him who is to come afterwards to uncover and extract the remaining riches of the ground. A wise administration of such a country will think it their duty to leave a just share of this inheritance to their descendants, who are entitled to expect it as the last legacies. National plans of cultivation should be formed on this principle, that the steps taken by the present cultivators for realizing part of the riches of the infant country shall not obstruct the works which will afterwards be necessary for also obtaining the remainder. This is carefully attended to in Holland and in China. No man is allowed to conduct the drains, by which he recovers a piece of marsh, in such a way as to render it much more difficult for a neighbour, or even for his own successor, to drain another piece, although it may at present be quite inaccessible. There remain in the middle of the most cultivated countries many marshes, which industry has not yet attempted to drain, and where the legislature has not been at pains to prevent many little abuses which have produced elevations in the beds of rivers, and rendered the complete draining of some spots impossible. Administration should attend to such things, because their consequences are great. The sciences and arts, by which alone these difficult and costly jobs can be performed, should be protected, encouraged, and cherished. It is only from science that we can obtain principles to direct these arts. The problem of draining canals is one of the most important, and yet has hardly ever occupied the attention of the hydraulic specialist. We apprehend that M. Boatt's theory will throw great light on it; and regret that the very limited condition of our present work will hardly afford room for a slight sketch of what may be done on the subject. We shall, however, attempt it by a general problem, which will involve most of the chief circumstances which occur in works of that kind.

**Quest. 6.** Let the hollow ground \( A \) (fig. 2.) be inundated by rains or springs, and have no outlet but the canal \( AB \), by which it discharges its water into the neighbouring river \( BCDE \), and that its surface is nearly on a level with that of the river at \( D \). It can only drain when the river sinks in the droughts of summer; and even if it could then drain completely, the putrid marsh would only be an infecting neighbour. It may be proposed to drain it by one or more canals; and it is required to determine their lengths and other dimensions, so as to produce the best effects?

It is evident that there are many circumstances to determine the choice, and many conditions to be attended to.

If the canals \( AC, AD, AG \), are respectively equal to the proportions \( BC, BD, BE \), of the river, and have the same slopes, they will have the same discharge; but they are not for this reason equivalent. The long canal \( AE \) may drain the marsh completely, while the short one AC will only do it in part; because the difference of level between A and C is but inconsiderable. Also the freshes of the river may totally obstruct the operation of AC, while the canal AE cannot be hurt by them, E being so much lower than C. Therefore the canal must be carried so far down the river, that no freshes there shall ever raise the waters in the canal so high as to reduce the slope in the upper part of it to such a level that the current shall not be sufficient to carry off the ordinary produce of water in the marsh.

Still the problem is indeterminate, admitting many solutions. This requisite discharge may be accomplished by a short but wide canal, or by a longer and narrower. Let us first see what solution can be made, so as to accomplish our purpose in the most economical manner, that is, by means of the smallest equation.—We shall give the solution in the form of an example.

Suppose that the daily produce of rains and springs raises the water 1½ inch on an area of a square league, which gives about 123,000 cubic fathoms of water. Let the bottom of the basin be three feet below the surface of the freshes in the river at B in winter. Also, that the slope of the river is 2 inches in 100 fathoms, or 1/50 depth, and that the canal is to be 6 feet deep.

The canal being supposed nearly parallel to the river, it must be at least 1800 fathoms long before it can be admitted into the river, otherwise the bottom of the bog will be lower than the mouth of the canal; and even then a hundred or two more fathoms added to this will give it so little slope, that an immense breadth will be necessary to make the discharge with so small a velocity. On the other hand, if the slope of the canal be made equal to that of the river, an extravagant length will be necessary before its admission into the river, and many obstacles may then intervene. And even then it must have a breadth of 13 feet, as may easily be calculated by the general hydraulic theorem. By receding from each of these extremes, we shall diminish the expense of excavation. Therefore,

Let x and y be the breadth and length, and h the depth (6 feet), of the canal. Let q be the depth of the bog below the surface of the river, opposite to the basin, D the discharge in a second, and \( \frac{1}{a} \) the slope of the river. We must make \( h \times y \) a minimum, or \( xy + y = 0 \).

The general formula gives the velocity

\[ V = \sqrt{\frac{N g d}{S}} \]

making \( \sqrt{\frac{N g}{S}} \) nearly 2 y b. This will be sufficiently exact for all cases which do not deviate far from this, because the velocities are very nearly in the subduplicate ratio of the slopes.

To introduce these data into the equation, recollect that \( V = \frac{D}{h x} \); \( d = \frac{h x}{x + 2 h} \). As to S, recollect that the canal being supposed of nearly equal length with the river, \( \frac{y}{a} \) will express the whole difference of height, and \( \frac{y}{a} - q \) is the difference of height for the canal. This quantity being divided by y, gives the value of

\[ \frac{1}{S} = \frac{y}{a} - q \]

Therefore the equation for the canal becomes

\[ \sqrt{\frac{N g}{S}} \left( \frac{h x}{x + 2 h} \right) \left( \frac{y}{a} - q \right) \]

Hence we deduce

\[ \frac{N g h^3 x^3}{a} - D(x + 2 h) \text{ and } y = \frac{3 N g h^3 x^3}{a} - D(x + 2 h) \]

If we substitute these values in the equation \( y + x = 0 \), and reduce it, we obtain finally,

\[ \frac{N g h^3 x^3}{a} - 3 x = 8 h \]

If we resolve this equation by making \( N g = (296)^2 \), or 87616 inches; \( h = 72 \), \( \frac{1}{a} = \frac{1}{7600} \), and \( D = 518400 \), we obtain \( x = 392 \) inches, or 32 feet 8 inches, and \( \frac{D}{h x} \) or \( V = 18.36 \) inches. Now putting these values in the exact formula for the velocity, we obtain the slope of the canal, which is \( \frac{1}{100} \), nearly 0.62 inches in 100 fathoms.

Let \( l \) be the length of the canal in fathoms. As the river has 2 inches fall in 100 fathoms, the whole fall is \( \frac{2}{100} \), and that of the canal is \( \frac{0.62}{100} \). The difference of these two must be 3 feet, which is the difference between the river and the entry of the canal. We have therefore \( \left( \frac{2 - 0.62}{100} \right) l = 36 \) inches. Hence \( l = 2664 \) fathoms; and this multiplied by the section of the canal gives 14177 cubic fathoms of earth to be removed.

This may surely be done, in most cases, for eight shillings each cubic fathom, which does not amount to 6000l., a very moderate sum for completely draining of nine square miles of country.

In order to judge of the importance of this problem, we have added two other canals, one longer and the other shorter, having their widths and slopes so adjusted as to ensure the same performance.

| Width | Velocity | Slope | Length | Excavation | |-------|----------|-------|--------|------------| | Feet | Inches | | | | | 42 | 14.28 | 1 | 2221 | 15547 | | 32\(\frac{1}{2}\) | 18.36 | 1 | 2604 | 14177 | | 21 | 28.57 | 1 | 7381 | 13933 |

We have considered this important problem in its most simple state. If the basin is far from the river, so that the drains are not nearly parallel to it, and therefore have less slope attainable in their course, it is more difficult. Perhaps the best method is to try two very extreme cases and a middle one, and then a fourth, nearer to that extreme which differs least from the middle one in the quantity of excavation. This will point out on which side the minimum of excavation lies, and also the law by which it diminishes and afterwards increases. Then draw a line, on which set off from one end the lengths of the canals. At each length erect an ordinate representing the excavation; and draw a regular curve through the extremities of the ordinates. From that point of the curve which is nearest to the base line, draw another ordinate to the base. This will point out the best length of the canal with sufficient accuracy. The length will determine the slope, and this will give the width, by means of the general theorem. N.B. These draining canals must always come off from the basin with evasated entries. This will prevent the loss of much fall at the entry.

Two canals may sometimes be necessary. In this case expense may frequently be saved, by making one canal flow into the other. This, however, must be at such a distance from the basin, that the swell produced in the other by this addition may not reach back to the immediate neighbourhood of the basin, otherwise it would impede the performance of both. For this purpose, recourse must be had to Problem III. in No. 104. of the article River. We must here observe, that in this respect canals differ exceedingly from rivers; rivers enlarge their beds, so as always to convey every increase of waters; but a canal may be gorged through its whole length, and will then greatly diminish its discharge. In order that the lower extremity of a canal may convey the waters of an equal canal admitted into it, their junction must be so far from the basin that the swell occasioned by raising its waters nearly ½ more (viz. in the subduplicate ratio of 1 to 2) may not reach back to the basin.

This observation points out another method of economy. Instead of one wide canal, we may make a narrower one of the whole length, and another narrow one reaching part of the way, and communicating with the long canal at a proper distance from the basin. But the lower extremity will now be too shallow to convey the waters of both. Therefore raise its banks by using the earth taken from its bed, which must at any rate be disposed of. Thus the waters will be conveyed, and the expense, even of the lower part of the long canal, will scarcely be increased.

These observations must suffice for an account of the management of open canals; and we proceed to the consideration of the conduct of water in pipes.

This is much more simple and regular, and the general theorem requires very trifling modifications for adapting it to the cases or questions that occur in the practice of the civil engineer. Pipes are always made round, and therefore \(d\) is always \(\frac{1}{4}\)th of the diameter. The velocity of water in a pipe which is in train, is

\[ V = \frac{307 (\sqrt{d} - 0.1)}{\sqrt{s - L/\sqrt{s + 1.6}}} = 0.3 (\sqrt{d} - 0.1) \text{ or } (\sqrt{d} - 0.1) \left( \frac{307}{\sqrt{s - L/\sqrt{s + 1.6}}} \right) \]

The chief questions are the following:

**Quest. 1.** Given the height \(H\) of the reservoir above the place of delivery, and the diameter and length of the pipe, to find the quantity of water discharged in a second.

Let \(L\) be the length, and \(A\) the fall which would produce the velocity with which the water enters the pipe, and actually flows in it, after overcoming all obstructions. This may be expressed in terms of the velocity by

\[ \frac{V^2}{2G}, \quad G \text{ denoting the acceleration of gravity, corresponding to the manner of entry. When no methods are adopted for facilitating the entry of the water, by a bell-shaped funnel or otherwise, } 2G \text{ may be assumed as } = 500 \text{ inches, or } 42 \text{ feet, according as we measure the velocity in inches or feet. The slope is } \frac{1}{s} = \frac{H}{2G}, \text{ which must be put into the general formula.} \]

This would make it very complicated. We may simplify it by the consideration that the velocity is very small in comparison of that arising from the height \(H\); consequently \(A\) is very small. Also, in the same pipe, the resistances are nearly in the duplicate ratio of the velocities when these are small, and when they differ little among themselves. Therefore make \(b = \frac{L}{h}\), taking \(h\) by guess, a very little less than \(H\). Then compute the mean velocity \(v\) corresponding to these data, or take it from the table. If \(h + \frac{v^2}{2G} = H\), we have found the mean velocity \(V = v\). If not, make the following proportion:

\[ h : \frac{v^2}{2G} = H : \frac{V^2}{2G}, \quad \text{which is the same with this, } h + \frac{v^2}{2G} : v^2 = H : V^2, \quad \text{and } V^2 = \frac{v^2 H}{h + \frac{v^2}{2G}} = \frac{v^2 H}{2Gh + v^2} = \frac{v^2 \cdot 2GH}{2G} \]

If the pipe has any bendings, they must be calculated for in the manner mentioned in the article River, No. 101; and the head of water necessary for overcoming this additional resistance being called \(\frac{V^2}{m}\), the last proportion must be changed for

\[ h + \frac{v^2}{2G} \left( \frac{1}{2G} + \frac{1}{m} \right) : v^2 = H : V^2 \]

**Quest. 2.** Given the height of the reservoir, the length of the pipe, and the quantity of water which is to be drawn off in a second; to find the diameter of the pipe which will draw it off?

Let \(d\) be considered as \(\frac{1}{4}\)th of the diameter, and let \(i : c\) represent the ratio of the diameter of a circle to its circumference. The section of the pipe is \(4c d^2\). Let the quantity of water per second be \(Q\); then \(\frac{Q}{4c d^2}\) is the mean velocity. Divide the length of the pipe by the height of the reservoir above the place of delivery, diminished by a very small quantity, and call the quotient \(S\). Consider this as the slope of the conduit; the general formula now becomes

\[ \frac{Q}{4c d^2} = \frac{307 (\sqrt{d} - 0.1)}{\sqrt{s - L/\sqrt{s + 1.6}}} = 0.3 (\sqrt{d} - 0.1), \quad \text{or } \] Water-works.

\[ Q = \frac{307(\sqrt{d} - 0.1)}{\sqrt{8}} - 0.3(\sqrt{d} - 0.1). \] We may neglect the last term in every case of civil practice, and also the small quantity \(0.1\). This gives the very simple formula,

\[ \frac{Q}{4c d^2} = \frac{307 \sqrt{d}}{\sqrt{8}} \]

from which we readily deduce

\[ d = \frac{Q \sqrt{8}}{4c \times 307} = \frac{Q \sqrt{8}}{3858}. \]

This process gives the diameter somewhat too small. But we easily rectify this error by computing the quantity delivered by the pipe, which will differ a little from the quantity proposed. Then observing, by this equation, that two pipes having the same length and the same slope give quantities of water, of which the squares are nearly as the fifth powers of the diameter, we form a new diameter in this proportion, which will be almost perfectly exact.

It may be observed that the height assumed for determining the slope in these two questions will seldom differ more than an inch or two from the whole height of the reservoir above the place of delivery; for in conduits of a few hundred feet long, the velocity seldom exceeds four feet per second, which requires only a head of three inches.

As no inconvenience worth minding results from making the pipes a tenth of an inch or so wider than is barely sufficient, and as this generally is more than the error arising from even a very erroneous assumption of \(h\), the answer first obtained may be augmented by one or two-tenths of an inch, and then we may be confident that our conduit will draw off the intended quantity of water.

We presume that every person who assumes the name of engineer knows how to reduce the quantity of water measured in gallons, pints, or other denominations, to cubic inches, and can calculate the gallons, &c., furnished by a pipe of known diameter, moving with a velocity that is measured in inches per second. We farther suppose that all care is taken in the construction of the conduit, to avoid obstructions occasioned by lumps of solder hanging in the inside of the pipes; and, particularly, that all the cocks and plugs by the way have waterways equal to the section of the pipe. Undertakers are most tempted to fail here, by making the cocks too small, because large cocks are very costly. But the employer should be scrupulously attentive to this; because a simple contraction of this kind may be the throwing away of many hundreds pounds in a wide pipe, which yields no more water than can pass through the small cock.

The chief obstructions arise from the deposition of sand or mud in the lower parts of pipes, or the collection of air in the upper parts of their bendings. The velocity being always very moderate, such depositions of heavy matters are unavoidable. The utmost care should therefore be taken to have the water freed from all such things at its entry by proper filtration; and there ought to be cleansing plugs at the lower parts of the bendings, or rather a very little way beyond them. When these are opened, the water issues with greater velocity, and carries the depositions with it.

It is much more difficult to get rid of the air which chokes the pipes by lodging in their upper parts. This air is sometimes taken in along with the water at the reservoir, when the entry of the pipe is too near the surface. This should be carefully avoided, and it costs no trouble to do so. If the entry of the pipe is two feet under the surface, no air can ever get in. Floats should be placed above the entries, having lids hanging from them, which will shut the pipe before the water runs too low.

But air is also disengaged from spring-water by merely passing along the pipe. When pipes are supplied by an engine, air is very often drawn in by the pumps in a disengaged state. It is also disengaged from its state of chemical union, when the pumps have a suction-pipe of 10 or 12 feet, which is very common. In whatever way it is introduced, it collects in all the upper part of bendings, and chokes the passage, so that sometimes not a drop of water is delivered. Our cocks should be placed there, which should be opened frequently by persons who have this in charge. Desaguliers describes a contrivance to be placed on all such eminences, which does this of itself. It is a pipe with a cock, terminating in a small cistern. The key of the cock has a hollow ball of copper at the end of a lever. When there is no air in the main pipe, water comes out by this discharger, fills the cistern, raises the ball, and thus shuts the cock. But when the bend of the main contains air, it rises into the cistern, and occupies the upper part of it. Thus the floating ball falls down, the cock opens and lets out the air, and the cistern again filling with water, the ball rises, and the cock is again shut.

A very neat contrivance for this purpose was invented by the late Professor Russel of Edinburgh. The cylindrical pipe BCDE (fig. 3.), at the upper end of a bending of the main, is screwed on, the upper end of which is a flat plate perforated with a small hole F. This pipe contains a hollow copper cylinder G, to the upper part of which is fastened a piece of soft leather H. When there is air in the pipe, it comes out by the hole A, and occupies the discharger, and then escapes through the hole F. The water follows, and, rising in the discharger, lifts up the hollow cylinder G, causing the leather H to apply itself to the plate CD, and shut the hole. Thus the air is discharged without the smallest loss of water.

It is of the most material consequence that there be no contraction in any part of a conduit. This is evident; but it is also prudent to avoid all unnecessary enlargements. For when the conduit is full of water moving along it, the velocity in every section is inversely proportional to the area of the section: it is therefore diminished wherever the pipe is enlarged; but it must again be increased where the pipe contracts. This cannot be without expending force in the acceleration. This consumes part of the impelling power, whether this be a head of water, or the force of an engine. See what is said on this subject in the article PUMPS, No. 83, &c. Nothing is gained by any enlargement; and every contraction, by requiring an augmentation of velocity, employs a part of the impelling force precisely equal to the weight of a column of water whose base is the contracted passage, and whose height is the fall which would produce a velocity equal to this augmentation. This point seems to have been quite overlooked by engineers of the first eminence, and has in many instances greatly greatly diminished the performance of their best works.

It is no less detrimental in open canals; because at every contraction a small fall is required for restoring the velocity lost in the enlargement of the canal, by which the general slope and velocity are diminished.

Another point which must be attended to in the conducting of water is, that the motion should not be sub- sultory, but continuous. When the water is to be driven along a main by the stroke of a reciprocating engine, it should be forced into an air-box, the spring of which may preserve it in motion along the whole subsequent main. If the water is brought to rest at every successive stroke of the piston, the whole mass must again be put in motion through the whole length of the main. This requires the same useless expenditure of power as to communicate this motion to as much dead matter; and this is over and above the force which may be necessary for raising the water to a certain height; which is the only circumstance that enters into the calculation of the power of the pump-engine.

An air-box removes this imperfection, because it keeps up the motion during the returning stroke of the piston. The compression of the air by the active stroke of the piston must be such as to continue the impulse in opposition to the contrary pressure of the water (if it is to be raised to some height), and in opposition to the friction or other resistances which arise from the motion that the water really acquires. Indeed a very considerable force is employed here also in changing the motion of the water, which is forced out of the capacious air-box into the narrow pipe; and when this change of motion is not judiciously managed, the expenditure of power may be as great as if all were brought to rest and again put in motion. It may even be greater, by causing the water to move in the opposite direction to its former motion. Of such consequence is it to have all these circumstances scientifically considered. It is in such particulars, unheeded by the ordinary herd of engineers or pump-makers, that the superiority of an intelligent practitioner is to be seen.

Another material point in the conduct of water in pipes is the distribution of it to the different persons who have occasion for it. This is rarely done from the rising main. It is usual to send the whole into a cistern, from which it is afterwards conducted to different places in separate pipes. Till the discovery of the general theorem by the chevalier Buat, this has been done with great inaccuracy. Engineers think that the different purchasers from water-works receive in proportion to their respective bargains when they give them pipes whose areas are proportional to these payments. But we now see, that when these pipes are of any considerable length, the waters of a larger pipe run with a greater velocity than those of a smaller pipe having the same slope. A pipe of two inches diameter will give much more water than four pipes of one inch diameter; it will give as much as five and a half such pipes, or more; because the squares of the discharges are very nearly as the fifth powers of the diameters. This point ought therefore to be carefully considered in the bargains made with the proprietors of water-works, and the payments made in this proportion. Perhaps the most unexceptionable method would be to make a double distribution. Let the water be first let off in its proper proportions into a second series of small cisterns, and let each have a pipe which will convey the whole water that is discharged into it. The first distribution may be made entirely by pipes of one inch in diameter; this would leave nothing to the calculation of the distributor, for every man would pay in proportion to the number of such pipes which run into his own cistern.

In many cases, however, water is distributed by pipes derived from a main. And here another circumstance comes into action. When water is passing along a pipe, its pressure on the sides of the pipe is diminished by its velocity; and if a pipe is now derived from it, the quantity drawn off is also diminished in the subduplicate ratio of the pressures. If the pressure is reduced to one-fourth, one-ninth, one-sixteenth, &c., the discharge from the lateral pipe is reduced to one-half, one-third, one-fourth, &c.

It is therefore of great importance to determine, what this diminution of pressure is which arises from the motion along the main.

It is plain, that if the water suffered no resistance in the main, its velocity would be that with which it entered, and it would pass along without exerting any pressure. If the pipe were shut at the end, the pressure on the sides would be the full pressure of the head of water. If the head of water remain the same, and the end of the tube be contracted, but not stopped entirely, the velocity in the pipe is diminished. If we would have the velocity in the pipe with this contracted mouth augmented to what it was before the contraction was made, we must employ the pressure of a piston, or of a head of water. This is propagated through the fluid, and thus a pressure is immediately excited on the sides of the pipe. New obstructions of any kind, arising from friction or any other cause, produce a diminution of velocity in the pipe. But when the natural velocity is checked, the particles react on what obstructs their motion; and this action is uniformly propagated through a perfect fluid in every direction. The resistance therefore which we thus ascribe to friction, produces the same lateral pressure, which a contraction of the orifice, which equally diminishes the velocity in the pipe, would do. Indeed this is demonstrable from any distinct notions that we can form of these obstructions. They proceed from the want of perfect smoothness, which obliges the particles next the sides to move in undulated lines. This excites transverse forces in the same manner as any constrained curvilinear motion. A particle in its undulated path tends to escape from it, and acts on the lateral particles in the same manner that it would do if moving singly in a capillary tube having the same undulations; it would press on the concave side of every such undulation. Thus a pressure is exerted among the particles, which is propagated to the sides of the pipe; or the diminution of velocity may arise from a viscidity or want of perfect fluidity. This obliges the particle immediately pressed to drag along with it another particle which is withheld by adhesion to the sides. This requires additional pressure from a piston, or an additional head of water; and this pressure also is propagated to the sides of the pipe.

Hence it should follow, that the pressure which water in motion exerts on the sides of its conduit is equal to that which is competent to the head of water which impels. impels it into the pipe, diminished by the head of water competent to the actual velocity with which it moves along the pipe. Let $H$ represent the head of water which impels it into the entry of the pipe, and $h$ the head which would produce the actual velocity; then $H-h$ is the column which would produce the pressure exerted on its sides.

This is abundantly verified by very simple experiments. Let an upright pipe be inserted into the side of the main-pipe. When the water runs out by the mouth of the main, it will rise in this branch till the weight of the column balances the pressure that supports it; and if we then ascertain the velocity of the issuing water by means of the quantity discharged, and compute the head or height necessary for producing this velocity, and subtract this from the height of water above the entry of the main, we shall find the height in the branch precisely equal to their difference. Our readers may see this by examining the experiments related by Gravesande, and still better by consulting the experiments narrated by Bossut, § 558, which are detailed with great minuteness; the results corresponded accurately with this proposition. The experiments indeed were not heights of water supported by this pressure, but water expelled by it through the same orifice. Indeed the truth of the proposition appears in every way we can consider the motion of water. And as it is of the first importance in the practice of conducting water (for reasons which will presently appear), it merits a particular attention. When an inclined tube is in train, the accelerating power of the water (or its weight diminished in the proportion of the length of the oblique column to its vertical height, or its weight multiplied by the fraction $\frac{1}{3}$, which expresses the slope), is in equilibrio with the obstructions; and therefore it exerts no pressure on the pipe but what arises from its weight alone. Any part of it would continue to slide down the inclined plane with a constant velocity, though detached from what follows it. It therefore derives no pressure from the head of water which impelled it into the pipe. The same must be said of a horizontal pipe infinitely smooth, or opposing no resistance. The water would move in this pipe with the full velocity due to the head of water which impels it into the entry. But when the pipe opposes an obstruction, the head of water is greater than that which would impel it into the pipe with the velocity that it actually has in it; and this additional pressure is propagated along the pipe, where it is balanced by the actual resistance, and therefore excites a quagga versum pressure on the pipe. In short, whatever part of the head of water in the reservoir, or of the pressure which impels it along the tube, is not employed in producing velocity, is employed in acting against some obstruction, and excites (by the reaction of this obstruction) an equal pressure on the tube. The rule therefore is general, but is subject to some modifications which deserve our attention.

In the simply inclined pipe BC (fig. 4.) the pressure on any point $S$ is equal to that of the head $AB$ of water which impels the water into the pipe, wanting or minus that of the head of water which would communicate to it the velocity with which it actually moves. This we shall call $x$, and consider it as the weight of a column of water whose length also is $x$. In like manner $H$ may be the column $AB$, which impels the water into the pipe, and would communicate a certain velocity; and $h$ may represent the column which would communicate the actual velocity. We have therefore $x = H-h$.

In the pipe HIKL, the pressure at the point $I$ is $AH-h-IO$, $=H-h-IO$; and the pressure at $K$ is $H-h+PK$.

And in the pipe DEFG, the pressure on $E$ is $AR-h-EM$, $=H-h-EM$; and the pressure at $F$ is $H-h+FN$.

We must carefully distinguish this pressure on any square inch of the pipe from the obstruction or resistance which that inch actually exerts, and which is part of the cause of this pressure. The pressure is (by the laws of hydrostatics) the same with that exerted on the water by a square inch of the piston or forcing head of water. This must balance the united obstructions of the whole pipe, in as far as they are not balanced by the relative weight of the water in an enclosed pipe. Whatever be the inclination of a pipe, and the velocity of the water in it, there is a certain part of this resistance which may not be balanced by the tendency which the water has to slide along it, provided the pipe be long enough; or if the pipe is too short, the tendency down the pipe may more than balance all the resistances that obtain below. In the first case, this surplus must be balanced by an additional head of water; and in the latter case the pipe is not in train, and the water will accelerate. There is something in the mechanism of these motions which makes a certain length of pipe necessary for bringing it into train; a certain portion of the surface which acts in concert in obstructing the motion. We do not completely understand this circumstance, but we can form a pretty distinct notion of its mode of acting. The film of water contiguous to the pipe is withheld by the obstruction, but glides along; the film immediately within this is withheld by the outer film, but glides through it; and thus all the concentric films glide within those around them, somewhat like the sliding tubes of a spyglass, when we draw it out by taking hold of the end of the innermost. Thus the second film passes beyond the first or outermost, and becomes the outermost, and rubs along the tube. The third does the same in its turn; and thus the central filaments come at last to the outside, and all sustain their greatest possible obstruction. When this is accomplished, the pipe is in train. This requires a certain length, which we cannot determine by theory. We see, however, that pipes of greater diameter must require a greater length, and this in a proportion which is probably that of the number of filaments, or the square of the diameter. Buat found this supposition agree well enough with his experiments. A pipe of one inch in diameter sustained no change of velocity by gradually shortening it till he reduced it to six feet, and then it discharged a little more water. A pipe of two inches diameter gave a sensible augmentation of velocity when shortened to 25 feet. He therefore says, that the square of the diameter in inches, multiplied by 72, will express (in inches) the length necessary for putting any pipe in train.

The resistance exerted by a square inch of the pipe makes but a small part of the pressure which the whole resistances occasion to be exerted there before they can be overcome. The resistance may be represented by \( \frac{d}{s} \), when \( d \) is the hydraulic depth (one-fourth of the diameter), and \( s \) the length of a column whose vertical height is one inch, and it is the relative weight of a column of water whose base is a square inch, and height is \( d \). For the resistance of any length \( s \) of pipe which is in train, is equal to the tendency of the water to slide down (being balanced by it); that is, is equal to the weight of this column multiplied by \( \frac{1}{s} \). The magnitude of this column is had by multiplying its length by its section. The section is the product of the border \( h \) or circumference, multiplied by the mean depth \( d \), or it is \( b \times d \). This multiplied by the length, is \( b \times d \times s \); and this multiplied by the slope \( \frac{1}{s} \) is \( b \times d \times s \), the relative weight of the column whose length is \( s \). The relative weight of one inch is therefore \( \frac{b \times d}{s} \); and this is in equilibrium with the resistance of a ring of the pipe one inch broad. This, when unfolded, is a parallelogram \( b \) inches in length. One inch of this therefore is \( \frac{d}{s} \), the relative weight of a column of water having \( d \) for its height and a square inch for its base. Suppose the pipe four inches in diameter, and the slope \( = 253 \), the resistance is one grain; for an inch of water weighs 253 grains.

This knowledge of the pressure of water in motion is of great importance. In the management of rivers and canals it instructs us concerning the damages which they produce in their beds by tearing up the soil; it informs us of the strength which we must give to the banks; but it is of more consequence in the management of close conduits. By this we must regulate the strength of our pipes; by this also we must ascertain the quantities of water which may be drawn off by lateral branches from any main conduit.

With respect to the first of these objects, where security is our sole concern, it is proper to consider the pressure in the most unfavourable circumstances, viz. when the end of the main is shut. This case is not unfrequent. Nay, when the water is in motion, its velocity in a conduit seldom exceeds a very few feet in a second. Eight feet per second requires only one foot of water to produce it. We should therefore estimate the strain on all conduits by the whole height of the reservoir.

In order to adjust the strength of a pipe to the strain, we may conceive it as consisting of two half cylinders of insuperable strength, joined along the two seams, where the strength is the same with the ordinary strength of the materials of which it is made. The inside pressure tends to burst the pipe by tearing open these seams; and each of these two seams is equal to the weight of a column of water whose height is the depth of the seam below the surface of the reservoir, and whose base is an inch broad and a diameter of the pipe in length. This follows from the common principles of hydrostatics.

Suppose the pipe to be of lead, one foot in diameter and 100 feet under the surface of the reservoir. Water weighs 62\(\frac{1}{2}\) pounds per foot. The base of our column is therefore \( \frac{1}{4} \) th of a foot, and the tendency to burst the pipe is \( 100 \times 62\frac{1}{2} \times \frac{1}{4} \text{th} = 62\frac{1}{2} \times 25 = 1575 \) pounds nearly. Therefore an inch of one seam is strained by 260 pounds. A rod of lead one inch square is pulled asunder by 860 pounds, (see Strength of Materials, No. 40.) Therefore if the thickness of the seam is \( \frac{1}{3} \) inches, or one-third of an inch, it will just withstand this strain. But we must make it much stronger than this, especially if the pipe leads from an engine which sends the water along it by starts. Belidor and Desaguliers have given tables of the thickness and weights of pipes which experience has found sufficient for the different materials and depths. Desaguliers says, that a leaden pipe of three-fourths of an inch in thickness is strong enough for a height of 140 feet and diameter of seven inches. From this we may calculate all others. Belidor says, that a leaden pipe 12 inches diameter and 60 feet deep should be half an inch thick; but these things will be more properly computed by means of the list given in No. 40 of the article Strength of Materials.

The application which we are most anxious to make of the knowledge of the pressure of moving waters is the derivation from a main conduit by lateral branches. This occurs very frequently in the distribution of waters among the inhabitants of towns; and it is so imperfectly understood by the greatest part of those who take the name of engineers, that individuals have no security that they shall get even one half of the water they bargain and pay for; yet this may be as accurately ascertained as any other problem in hydraulics by means of our general theorem. The case therefore merits our particular attention.

It appears to be determined already, when we have ascertained the pressures by which the water is impelled into these lateral pipes, especially after we have said that the experiments of Bossut on the actual discharges from a lateral pipe fully confirm the theoretical doctrine. But much remains to be considered. We have seen that there is a vast difference between the discharge made through a hole, or even through a short pipe, and the discharge from the far end of a pipe derived from a main conduit. And even when this has been ascertained by our new theory, the discharge thus modified will be found considerably different from the real state of things: For when water is flowing along a main with a known velocity, and therefore exerting a known pressure on the circle which we propose for the entry of a branch, if we insert a branch there water will go along it; but this will generally make a considerable change in the motion along the main, and therefore in the pressure which is to expel the water. It also makes a considerable change in the whole quantity which passes along the anterior part of the main, and a still greater change on what moves along that part of it which lies beyond the branch: it therefore affects the quantity necessary for the whole supply, the force that is required for propelling it, and the quantity delivered by other branches. This part therefore of the management of water in conduits is of considerable importance and intricacy. We can propose in this place nothing more than a solution of such leading questions as involve the chief circumstances, recommending to our readers the perusal of original works on this subject. M. Bossut's experiments experiments are fully competent to the establishment of the fundamental principle. The hole through which the lateral discharges were made was but a few feet from the reservoir. The pipe was successively lengthened, by which the resistances were increased, and the velocity diminished. But this did not affect the lateral discharges, except by affecting the pressures; and the discharges from the end of the main were supposed to be the same as when the lateral pipe was not inserted. Although this was not strictly true, the difference was insensible, because the lateral pipe had but about the 1/8th part of the area of the main.

Suppose that the discharge from the reservoir remains the same after the derivation of this branch, then the motion of the water all the way to the insertion of the branch is the same as before; but, beyond this, the discharge is diminished by all that is discharged by the branch, with the head x equivalent to the pressure on the side. The discharge by the lower end of the main being diminished, the velocity and resistance in it are also diminished. Therefore the difference between x and the head employed to overcome the friction in this second case, would be a needless or insufficient part of the whole load at the entry, which is impossible; for every force produces an effect, or it is destroyed by some reaction. The effect of the forcing head of water is to produce the greatest discharge corresponding to the obstructions; and thus the discharge from the reservoir, or the supply to the main, must be augmented by the insertion of the branch, if the forcing head of water remains the same. A greater portion therefore of the forcing head was employed in producing a greater discharge at the entry of the main, and the remainder, less than x, produced the pressure on the sides. This head was the one competent to the obstructions resulting from the velocity beyond the insertion of the branch; and this velocity, diminished by the discharge already made, was less than that at the entry, and even than that of the main without a branch. This will appear more distinctly by putting the case into the form of an equation. Therefore let H—x be the height due to the velocity at the entry, of which the effect obtains only horizontally. The head x is the only one which acts on the sides of the tube, tending to produce the discharge by the branch, at the same time that it must overcome the obstructions beyond the branch. If the orifice did not exist, and if the force producing the velocity on a short tube be represented by 2 G, and the section of the main be A, the supply at the entry of the main would be $A \sqrt{2G} \sqrt{H-x}$; and if the orifice had no influence on the value of x, the discharge by the orifice would be $D \sqrt{\frac{x}{H}}$, D being its discharge by means of the head H, when the end of the main is shut; for the discharges are in the subduplicate ratio of the heads of water by which they are expelled; and therefore

$$\sqrt{H} : \sqrt{x} = D : D \sqrt{\frac{x}{H}} (=\frac{3}{2}).$$

But we have seen that x must diminish; and we know that the obstructions are nearly as the square roots of the velocities, when these do not differ much among themselves. Therefore calling y the pressure or head which balances the resistances of the main without a branch, while x is the head necessary for the main with a branch, we may institute this proportion, $y : H-y = \frac{x(H-y)}{y}$; and this 4th term will express the head producing the velocity in the main beyond the branch (as H—y would have done in a main without a branch). This velocity beyond the branch will be $\sqrt{2G} \sqrt{\frac{x(H-y)}{y}}$, and the discharge at the end will be $A \sqrt{2G} \sqrt{\frac{x(H-y)}{y}}$. If to this we add the discharge of the branch, the sum will be the whole discharge, and therefore the whole supply. Therefore we have the following equation,

$$A \sqrt{2G} \sqrt{\frac{x(H-y)}{y}} + D \sqrt{\frac{x}{H}}.$$ From this we deduce the value of x

$$= \frac{2GHA^2}{(A \sqrt{2G} \sqrt{\frac{H-y}{y}} + \sqrt{\frac{x}{H}})^2 + 2GA^2}.$$ This value of x being substituted in the equation of the discharge of the branch, which was $D \sqrt{\frac{x}{H}}$, will give the discharges required, and they will differ so much the more from the discharges calculated according to the simple theory, as the velocity in the main is greater. By the simple theory, we mean the supposition that the lateral discharges are such as would be produced by the head $H-h$, where H is the height of the reservoir, and h the head due to the actual velocity in the main.

And thus it appears that the proportion of the discharge by a lateral pipe from a main that is shut at the far end, and the discharge from a main that is open, depends not only on the pressures, but also on the size of the lateral pipe, and its distance from the reservoir. When it is large, it greatly alters the train of the main, under the same head, by altering the discharge at its extremity, and the velocity in it beyond the branch; and if it be near the reservoir, it greatly alters the train, because the diminished velocity takes place through a greater extent, and there is a greater diminution of the resistances.

When the branch is taken off at a considerable distance from the reservoir, the problem becomes more complicated, and the head x is resolved into two parts; one of which balances the resistance in the first part of the main, and the other balances the resistances beyond the lateral pipe, with a velocity diminished by the discharge from the branch.—A branch at the end of the main produces very little change in the train of the pipe.

When the lateral discharge is great, the train may be so altered, that the remaining part of the main will not run full, and then the branch will not yield the same quantity. The velocity in a very long horizontal tube may be so small (by a small head of water and great obstructions in a very long tube) that it will just run full. An orifice made in its upper side will yield nothing; and yet a small tube inserted into it will carry a column almost as high as the reservoir. So that we cannot judge in all cases of the pressures by the discharges, and vice versa. If there be an inclined tube, having a head greater than what is competent to the velocity, we may bring it into train by an opening on its upper side near the reservoir. This will yield some water, and the velocity will diminish in the tube till it is in train. If we should now enlarge the hole, it will yield no more water than before.

And thus we have pointed out the chief circumstances which affect these lateral discharges. The discharges are afterwards modified by the conduits in which they are conveyed to their places of destination. These being generally of small dimensions, for the sake of economy, the velocity is much diminished. But, at the same time, it approaches nearer to that which the same conduit would bring directly from the reservoir, because its small velocity will produce a less change in the train of the main conduit.

We should now treat of jets of water, which still make an ornament in the magnificent pleasure grounds of the wealthy. Some of these are indeed grand objects, such as the two at Peterhoff in Russia, which spout about 60 feet high a column of nine inches diameter, which falls again, and shakes the ground with its blow. Even a spout of an inch or two inches diameter, lancing to the height of 150 feet, is a gay object, and greatly enlivens a pleasure ground; especially when the changes of a gentle breeze bend the jet to one side. But we have no room left for treating this subject, which is of some nicety; and must conclude this article with a very short account of the management of water as an active power for impelling machinery.

II. Of Machinery driven by Water.

This is a very comprehensive article, including almost every possible species of mill. It is no less important, and it is therefore matter of regret, that we cannot enter into the detail which it deserves. The mere description of the immense variety of mills which are in general use, would fill volumes, and a scientific description of their principles and maxims of construction would almost form a complete body of mechanical science. But this is far beyond the limits of a work like ours. Many of these machines have been already described under their proper names, or under the articles which give an account of their manufactures; and for others we must refer our readers to the original works, where they are described in minute detail. The great academical collection Des Arts et Metiers, published in Paris in many folio volumes, contains a description of the peculiar machinery of many mills; and the volumes of the Encyclopedie Methodique, which particularly relate to the mechanic arts, already contain many more. All that we can do in this place is, to consider the chief circumstances that are common to all water-mills, and from which all must derive their efficacy. These circumstances are to be found in the manner of employing water as an acting power, and most of them are comprehended in the construction of water-wheels. When we have explained the principles and the maxims of construction of a water-wheel, every reader conversant in mechanics knows, that the axis of this wheel may be employed to transmit the force impressed on it to any species of machinery. Therefore nothing subsequent to this can with propriety be considered as water-works.

Water-wheels are of two kinds, distinguished by the manner in which water is made an impelling power, viz. by its weight, or by its impulse. This requires a very different form and manner of adaptation; and this forms an ostensible distinction, sufficiently obvious to give a name to each class. When water is made to act by its weight, it is delivered from the spout as high on the wheel as possible, that it may continue long to press it down: but when it is made to strike the wheel, it is delivered as low as possible, that it may have previously acquired a great velocity. And thus the wheels are said to be OVERSHOT or UNDERSHOT.

Of Overshot Wheels.

This is nothing but a frame of open buckets, so disposed round the rim of a wheel as to receive the water delivered from a spout; so that one side of the wheel is loaded with water, while the other is empty. The consequence must be, that the loaded side must descend. By this motion the water runs out of the lower buckets, while the empty buckets of the rising side of the wheel come under the spout in their turn, and are filled with water.

If it were possible to construct the buckets in such a manner as to remain completely filled with water till they come to the very bottom of the wheel, the pressure with which the water urges the wheel round its axis would be the same as if the extremity of the horizontal radius were continually loaded with a quantity of water sufficient to fill a square pipe, whose section is equal to that of the bucket, and whose length is the diameter of the wheel. For let the buckets BD and EF (fig. 5.) Fig. 5. be compared together, the arches DB and EF are equal. The mechanical energy of the water contained in the bucket EF, or the pressure with which its weight urges the wheel, is the same as if all this water were hung on that point T of the horizontal arm CF, where it is cut by the vertical or plumb-line BT. This is plain from the most elementary principles of mechanics. Therefore the effect of the bucket BD is to that of the bucket EF as CT to CF or CB. Draw the horizontal lines PB b b, QD d d. It is plain, that if BD is taken very small, so that it may be considered as a straight line, BD : BO = CB : BP, and EF : b d = CF : CT, and EF × CT = b d × CE. Therefore if the prism of water, whose vertical section is b b d d, were hung on at F, its force to urge the wheel round would be the same as that of the water lying in the bucket BD. The same may be said of every bucket; and the effective pressure of the whole ring of water A/HKFI, in its natural situation, is the same with the pillar of water a h h a hung on at F. And the effect of any portion BF of this ring is the same with that of the corresponding portion b F f b of the vertical pillar. We do not take into account the small difference which arises from the depth B or F f, because we may suppose the circle described through the centres of gravity of the buckets. And in the farther prosecution of this subject, we shall take similar liberties, with the view of simplifying the subject, and saving time to the reader.

But such a state of the wheel is impossible. The bucket at the very top of the wheel may be completely filled with water; but when it comes into the oblique position BD, a part of the water must run over the outer edge 2, and the bucket will only retain the quan-