WATER-WORKS (1842) — Encyclopaedia Britannica Historical Editions

WATER-WORKS

Volume 21 · 29,285 words · 1842 Edition

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 classed 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 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 work, particularly in Hydrodynamics, part iii.—On Hydraulic Machinery. 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 along the axis of the water-wheel, this may be set to the performance of any task whatever.

CLASS I.

Of the Conducting of Water.

This is undoubtedly a business of great importance, and forms 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 with his work will convey, or what plan and dimensions of conduit will convey the quantity which may be proposed.

In the article River 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 we know in hydrostatics, seems to have been entirely ignorant of any principle 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 rivers. Its motion in pipes had been partially considered in detached portions 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 Abbé Bossut, narrated in his Hydrodynamique, are of the greatest value, having been made in the cases of most general frequency, and with great care. The greatest service, however, has been done by the Chevalier du 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 however necessary for connecting these experiments. The sufficiency of this principle was not easily ascertained. Du Buat's way of establishing it 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 on 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 du 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.

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 Table I. Logarithms of the Values of the Numerator of the Fraction $ \frac{3}{\sqrt{d} - 0.1} $ for every Value of the Hydraulic mean Depth d; also the Value of 0.3 $ (\sqrt{d} - 0.1) $.

| d | Log. of \( \frac{3}{\sqrt{d} - 0.1} |-----|--------------------------------------|----- | 0.1 | 1.82208 | 0.06 | 0.2 | 2.02773 | 0.10 | 0.3 | 2.13315 | 0.13 | 0.4 | 2.21343 | 0.16 | 0.5 | 2.27400 | 0.18 | 0.6 | 2.31618 | 0.20 | 0.7 | 2.35414 | 0.22 | 0.8 | 2.38719 | 0.24 | 0.9 | 2.41538 | 0.25 | 1.0 | 2.44138 | 0.27 | 1.1 | 2.46431 | 0.28 | 1.2 | 2.48518 | 0.30 | 1.3 | 2.50426 | 0.31 | 1.4 | 2.52135 | 0.32 | 1.5 | 2.53618 | 0.34 | 1.6 | 2.55035 | 0.35 | 1.7 | 2.56379 | 0.36 | 1.8 | 2.57612 | 0.37 | 1.9 | 2.59331 | 0.38 | 2.0 | 2.60539 | 0.39 | 2.1 | 2.61713 | 0.40 | 2.2 | 2.62829 | 0.41 | 2.3 | 2.63839 | 0.42 | 2.4 | 2.64827 | 0.44 | 2.5 | 2.65772 | 0.45 | 2.6 | 2.66631 | 0.45 | 2.7 | 2.67556 | 0.46 | 2.8 | 2.68395 | 0.47 | 2.9 | 2.69207 | 0.48 | 3.0 | 2.69969 | 0.49 | 3.1 | 2.70743 | 0.50 | 3.2 | 2.71472 | 0.51 | 3.3 | 2.72131 | 0.52 | 3.4 | 2.72760 | 0.53 | 3.5 | 2.73331 | 0.53 | 3.6 | 2.74178 | 0.54 | 3.7 | 2.74905 | 0.55 | 3.8 | 2.75417 | 0.56 | 3.9 | 2.76009 | 0.56 | 4.0 | 2.76589 | 0.57 | 4.1 | 2.77153 | 0.58 | 4.2 | 2.77704 | 0.59 | 4.3 | 2.78240 | 0.59 | 4.4 | 2.78765 | 0.60 | 4.5 | 2.79277 | 0.60 | 4.6 | 2.79779 | 0.61 | 4.7 | 2.80269 | 0.62 | 4.8 | 2.80747 | 0.63 **Table II. Logarithms of the Values of the \) | 0.3 $ (\sqrt{d} - 0.1) $ | d | Log. of $ \frac{3}{\sqrt{d} - 0.1} $ | 0.3 $ (\sqrt{d} - 0.1) $ | d | Log. of $ \frac{3}{\sqrt{d} - 0.1} $ | 0.3 $ (\sqrt{d} - 0.1) $ | ---------------------|-----|--------------------------------------|--------------------------|-----|--------------------------------------|--------------------------| | 4.9 | 2.41216 | 0.63 | 9.7 | 2.96634 | 0.90 | | 5.0 | 2.31674 | 0.63 | 9.8 | 2.96855 | 0.91 | | 5.1 | 2.23125 | 0.65 | 9.9 | 2.97093 | 0.91 | | 5.2 | 2.15267 | 0.65 | 10 | 2.97319 | 0.92 | | 5.3 | 2.08300 | 0.66 | 11 | 2.97544 | 0.92 | | 5.4 | 2.02342 | 0.67 | 12 | 3.01401 | 0.91 | | 5.5 | 1.96380 | 0.67 | 13 | 3.03129 | 0.91 | | 5.6 | 1.90428 | 0.68 | 14 | 3.04843 | 0.90 | | 5.7 | 1.84476 | 0.68 | 15 | 3.06383 | 0.91 | | 5.8 | 1.78524 | 0.69 | 16 | 3.07829 | 0.91 | | 5.9 | 1.72572 | 0.69 | 17 | 3.09170 | 0.91 | | 6.0 | 1.66620 | 0.70 | 18 | 3.10441 | 0.91 | | 6.1 | 1.60668 | 0.70 | 19 | 3.11644 | 0.91 | | 6.2 | 1.54716 | 0.71 | 20 | 3.12753 | 0.91 | | 6.3 | 1.48764 | 0.72 | 21 | 3.13867 | 0.91 | | 6.4 | 1.42812 | 0.72 | 22 | 3.14899 | 0.91 | | 6.5 | 1.36860 | 0.73 | 23 | 3.15855 | 0.91 | | 6.6 | 1.30908 | 0.73 | 24 | 3.16828 | 0.91 | | 6.7 | 1.24956 | 0.74 | 25 | 3.17734 | 0.91 | | 6.8 | 1.19004 | 0.74 | 26 | 3.18601 | 0.91 | | 6.9 | 1.13052 | 0.75 | 27 | 3.19433 | 0.91 | | 7.0 | 1.07100 | 0.76 | 28 | 3.20243 | 0.91 | | 7.1 | 1.01148 | 0.76 | 29 | 3.21020 | 0.91 | | 7.2 | 0.95196 | 0.77 | 30 | 3.21757 | 0.91 | | 7.3 | 0.89244 | 0.77 | 31 | 3.22465 | 0.91 | | 7.4 | 0.83292 | 0.78 | 32 | 3.23196 | 0.91 | | 7.5 | 0.77340 | 0.78 | 33 | 3.23877 | 0.91 | | 7.6 | 0.71388 | 0.79 | 34 | 3.24537 | 0.91 | | 7.7 | 0.65436 | 0.79 | 35 | 3.25176 | 0.91 | | 7.8 | 0.59484 | 0.80 | 36 | 3.25799 | 0.91 | | 7.9 | 0.53532 | 0.80 | 37 | 3.26404 | 0.91 | | 8.0 | 0.47580 | 0.81 | 38 | 3.26993 | 0.91 | | 8.1 | 0.41628 | 0.82 | 39 | 3.27566 | 0.91 | | 8.2 | 0.35676 | 0.82 | 40 | 3.28125 | 0.91 | | 8.3 | 0.29724 | 0.83 | 41 | 3.28669 | 0.91 | | 8.4 | 0.23772 | 0.84 | 42 | 3.29201 | 0.91 | | 8.5 | 0.17820 | 0.84 | 43 | 3.29729 | 0.91 | | 8.6 | 0.11868 | 0.85 | 44 | 3.30257 | 0.91 | | 8.7 | 0.05916 | 0.85 | 45 | 3.30772 | 0.91 | | 8.8 | 0.00000 | 0.86 | 46 | 3.31207 | 0.91 | | 8.9 | 0.00000 | 0.86 | 47 | 3.31638 | 0.91 | | 9.0 | 0.00000 | 0.86 | 48 | 3.32051 | 0.91 | | 9.1 | 0.00000 | 0.86 | 49 | 3.32454 | 0.91 | | 9.2 | 0.00000 | 0.86 | 50 | 3.32849 | 0.91 | | 9.3 | 0.00000 | 0.86 | 51 | 3.33240 | 0.91 | | 9.4 | 0.00000 | 0.86 | 52 | 3.33608 | 0.91 | | 9.5 | 0.00000 | 0.86 | 53 | 3.33968 | 0.91 | | 9.6 | 0.00000 | 0.86 | 54 | 3.34327 | 0.91 | Denominator of the Fraction**

\[ \frac{307 (\sqrt{d} - 0.1)}{\sqrt{s} - L\sqrt{s} + 1.6} \]

for every Value of the Slope $ s $.

| Log. of $ \sqrt{s} - L\sqrt{s} + 1.6 $ | $ s $ | |------------------------------------------|------| | 9.71784 | 7.3 | | 9.74210 | 7.4 | | 9.76638 | 7.5 | | 9.78376 | 7.6 | | 9.80262 | 7.7 | | 9.81829 | 7.8 | | 9.83461 | 7.9 | | 9.84930 | 8.0 | | 9.86314 | 8.1 | | 9.87622 | 8.2 | | 9.88857 | 8.3 | | 9.90031 | 8.4 | | 9.91153 | 8.5 | | 9.92227 | 8.6 | | 9.93247 | 8.7 | | 9.94231 | 8.8 | | 9.95173 | 8.9 | | 9.96055 | 9.0 | | 9.96962 | 9.1 | | 9.97818 | 9.2 | | 9.98632 | 9.3 | | 9.99427 | 9.4 | | 9.99200 | 9.5 | | 9.98945 | 9.6 | | 9.98639 | 9.7 | | 9.98273 | 9.8 | | 9.98064 | 9.9 | | 9.97833 | 10. |

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 in the article.

**Table I.** consists of three columns. Column 1, entitled "Log. of $ \sqrt{s} - L\sqrt{s} + 1.6 $", contains the hydraulic mean depths of any conduit in inches. This is set down for every tenth of an inch in the first column, so that the answers may be more accurately obtained for pipes, the mean depth of which seldom exceed three or four inches. The column is continued to 100 inches, which is fully equal to the hydraulic mean depth of any canal.

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 reciprocal of the number 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} - L\sqrt{s + 1} \cdot 6 $.

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 forty-four 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} $ or 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, or 20.08.

6. To obtain the Scottish pints per minute (each containing 103.4 cubic inches), multiply the velocity by 60, and this product by $ 5^3 $, 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. 5^3 or 25 .................................................. 1.39794 log. of 0.7854 .................................................... 1.89509

Subtract the log. of 103.4 ....................................... 2.01451

Remains the log. of 228.8 pints .................................. 2.35943

Example 2. The canal mentioned in the article River, p. 270, 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 5$\frac{1}{2}$ 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 = 15840 $. 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 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 Du Burt's, and comes nearer in these cases. Instead of taking the hyperbolic logarithm of $ \sqrt{s} + 1.6 $, multiply its common logarithm by 21, 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 the Chevalier du Bart's formula is chiefly founded on observations on small canals; and is therefore most accurate in 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 River.

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 twenty-five inches: there must be a velocity of thirty inches at the surface, as appears by the table in the article River, p. 273. 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 KC (fig. 1), till it comes near the entry of the canal AB, and there it acquires the 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 difficult, even in a rectangular canal. It was one great aim of Du Buat'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 du Buat. 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 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 $\sqrt{s - L \sqrt{s + 1}}$. $N$ is a number to be fixed by experiment (see River), 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 River, p. 265). Then with $\sqrt{d}$ and the slope find $V$ by the table: make this $V = \frac{\sqrt{N} g d}{\sqrt{S}}$. This gives $\sqrt{N} g = \frac{V \sqrt{S}}{\sqrt{d}}$. 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 velo- city in the canal will be $ \frac{\sqrt{Ng}}{\sqrt{S}} \left( \frac{w + 2h}{w} \right) $. The height which will produce this velocity is $ \frac{Ng}{2GS} \left( \frac{w + 2h}{w} \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

\[ h = \frac{-\left[ w \left( \frac{Ng}{2GS} + 1 \right) - 2H \right]}{4} \]

\[ + \frac{\sqrt{8Hw + \left[ w \left( \frac{Ng}{2GS} + 1 \right) - 2H \right]^2}}{4}. \]

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 depth $ 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} = \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{Ng}}{\sqrt{S}} \left( \frac{y + 2x}{y} \right) $.

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}{100} $. 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 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^2}{744} $ be taken for the height producing the velocity, it cannot exceed three inches, nor be less than one. Suppose it two inches, and therefore the depth of the stream in the canal to be sixteen inches; find the mean width of the canal by the equation $ w = \frac{Q}{h(\sqrt{d} - 0.1)(\sqrt{S} - 0.3)} $, in which $ Q $ is twenty cubic feet (the sixtieth part of 1200), $ \sqrt{S} $ is $ 28-153 = \sqrt{1000 - 1} \sqrt{1000 + 1} - 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 two inches. Take this from eighteen, and there remains 16-48, 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 the 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. de 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 two to five; for the waters have been brought in, and exceed the report in this proportion. Indeed, long after giving in the report, M. Perronot, the most celebrated engineer in France, affirmed that the dimensions proposed were much greater than were necessary, and said that an aqueduct of five and a half feet wide and three and a half deep, with a slope of fifteen inches in a thousand fathoms, would have a velocity of twelve or thirteen 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 nineteen inches when it was three and a half feet deep. M. Perronot 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. Perronot's aqueduct be examined by our general formula, $ s $ will be found $ = 16 $, and $ d = 18-72 $, from which we deduce the velocity $ = 18 $, 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 or 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 m-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, 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 a 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 its diminution, and the quantity discharged may be small. Improbable as this may appear, it is not demonstrably false; and hence we see the propriety of the following question:

**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 effective fall, and the velocity is $\sqrt{2G} \sqrt{H - x}$. This multiplied by $wx$ will give the quantity discharged. Therefore $wx\sqrt{2G} \sqrt{H - x}$ must be made a maximum. The common process for this will give the equation $2H = 3x + \frac{3}{2}H$. The mean velocity will be $\sqrt{2G} \sqrt{\frac{3}{2}H}$; the section will be $\frac{3}{2}wH$, and the discharge $= \sqrt{2G} wH \sqrt{\frac{3}{2}H}$, and $d = \frac{3}{2}wH$. With these data the slope easily had by the formula for uniform motion.

If the canal is of the trapezoidal form, the investigation is 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 things must happen; either the motion will not acquire sufficient velocity in such a canal for want of 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 becomes a simple weir or waste-board. Now the discharge of a waste-board is $\frac{3}{2} \sqrt{2G} w \left( \frac{3}{2} - \frac{1}{2}h \right)$. The maximum determined by the preceding problem is to that of a waste-board of the same dimensions as $H \sqrt{\frac{3}{2}H} = \frac{3}{2} \times 6465$, nearly $= 9 : 10$.

Having given the dimensions and slope of a canal, we can discover the relation between its expenditure and the time; we can tell how much it will sink the surface of a pond in twenty-four 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 $wh\sqrt{\frac{N_g}{S}}$ for the canal. These must be the same. This gives the equation $eb\sqrt{H-h}\sqrt{2G} = wh\sqrt{\frac{N_g}{S}}$, 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 River. 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 $eb\sqrt{H-h}\sqrt{2G} = wh\sqrt{\frac{N_g}{S}}$ gives $h = \frac{G eb^2 S}{w^2 N_g d} + \sqrt{\frac{G eb^2 S H}{w^2 N_g d} + \left( \frac{G eb^2 S}{w^2 N_g d} \right)^2}$. 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

$$eb = \frac{wh\sqrt{Ngd}}{\sqrt{2}GS(H-h)} = \frac{Q}{\sqrt{2}G(H-h)}$$

If the area of the sluice is known, and we would learn the height to which it will raise the river, we have $H-h = \frac{Q^2}{2Ge^2b}$, for the expression of the rise of the 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, $\frac{Q^2}{2Ge^2b}$, 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 object, 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 were not only useless, but hurtful by their 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 franchised 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 benevolent views, and will be careful not to proceed by wantonly snatching at present any partial good, and by picking out what is most easily procured, 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 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 the Chevalier du Batz'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 let its surface be nearly on a level with that of the river at B. It can only drain when the river sinks in the droughts of summer; and even if it could then drain completely, the 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

\[ \frac{D}{h x} = \sqrt{\frac{N g}{x + 2 h}} \cdot \frac{\sqrt{\frac{y}{a} - q}}{y}. \]

Hence we deduce $ y = \frac{N g q h^3 x^3}{a} - D^2 (x + 2 h) $

\[ dy = \frac{3 N g q h^3 x^2 d x}{a} - D^2 (x + 2 h) \]

If we substitute these values in the equation $ y dx + x dy = 0 $, and reduce it, we obtain finally

\[ \frac{N g h^3 x^3}{a D^2} - 3 x = 8 h. \]

If we resolve this equation by making $ N g = (296)^2 $, or 87616 inches, $ h = 72 $, $ \frac{1}{a} = \frac{1}{5400} $, 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 two inches fall in 100 fathoms, the whole fall is $ \frac{2}{100} $ and that of the canal is $ \frac{1}{100} $. The difference of these two must be three feet, which is the difference between the river and the entry of the canal. We have therefore

\[ \left( \frac{2}{100} - \frac{1}{100} \right) l = 36 \text{ inches}. \quad \text{Hence } l = 2604 \text{ fathoms}; \]

and this multiplied by the section of the canal gives 14,177 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 L6000, a very moderate sum for completely draining 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 insure the same performance.

| Width | Velocity | Slope | Length | Excavation | |-------|----------|-------|--------|------------| | Feet | Inches | | | | | 42 | 14-28 | 18-58 | 2221 | 15547 | | 32$\frac{3}{4}$ | 18-36 | 17-65 | 2604 | 14177 | | 21 | 28-57 | 17-81 | 7381 | 15833 |

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. 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. p. 283, 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

\[ V = \frac{307 (\sqrt{d} - 0.1)}{\sqrt{s} - L \sqrt{s} + 16} \]

$= 0.3 (\sqrt{d} - 0.1)$, or $= (\sqrt{d} - 0.1) \left( \frac{307}{\sqrt{s} - L \sqrt{s} + 16} \right)$.

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 $h$ 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

\[ \frac{V^2}{2G} \]

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, $2G$ may be assumed as $= 500$ inches, or 42 feet, according as we measure the velocity in inches or feet. The slope is

\[ \frac{1}{s} = \frac{H - 2G}{L} \]

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 $h$ 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 $s = \frac{L}{h}$, 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.

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

\[ h + \frac{v^2}{2G} : v^2 = H : V^2, \text{ and } V^2 = \frac{v^2 H}{h + \frac{v^2}{2G}} = \frac{v^2 H}{2G h + v^2} \]

\[ = \frac{v^2 \cdot 2GH}{v^2 + 2GH}. \]

If the pipe has any bendings, they must be calculated for in the manner mentioned in the article River; and the head of water necessary for overcoming this additional resistance being called $V^2$, the last proportion must be changed for

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

**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 $1 : e$ represent the ratio of the diameter of a circle to its circumference. The section of the pipe is $4ed^2$. Let the quantity of water per second be $Q$; then $\frac{Q}{4ed^2}$ 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

\[ \frac{Q}{4ed^2} = \frac{307 (\sqrt{d} - 0.1)}{\sqrt{s} - L \sqrt{s} + 16} = 0.3 (\sqrt{d} - 0.1), \text{ or } \]

\[ \frac{Q}{4ed^2} = \frac{(307 (\sqrt{d} - 0.1)}{\sqrt{s}} = 0.3 (\sqrt{d} - 0.1). \]

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

\[ \frac{Q}{4ed^2} = \frac{307 \sqrt{d}}{\sqrt{s}}, \]

from which we readily deduce

\[ d = \frac{Q \sqrt{s}}{4e \times 307} = \frac{Q \sqrt{s}}{3858}. \]

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. I 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.

No inconvenience worth minding results from making the pipe 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 further 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 throwing away many hundreds of 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 pits 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 deposition with it.

It is much more difficult to get rid of the air which chokes the pipes by lodging in their upper parts. The 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 ever gets in. Floats should be placed above the entries, having lids hanging from them, which will shut the pipe before the water runs too low.

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 ten or twelve feet, which is very common. In whatever way it is introduced, it collects in all the upper parts 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 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 the discharger, fills the cistern, raises the ball, and thus shuts the cock. But when the bend of the main contains air, 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 Russell 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, which 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 Pump. 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 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 subsultory, 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 also employed here 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 du Buat, this has been done with great inaccuracy. Engineers think that the different purchasers from waterworks 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 runs 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 re-act 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 viscosity 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 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 examining the experiments related by Bossut, which are detailed with great minuteness. The results correspond 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 $\frac{1}{2}$, 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 quinquiescent 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 re-action 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 of water AB which im- The water into the pipe, wanting or minus that of the head of water which would communicate to 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 $ AI - 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 - M = H - h - EM $; and the pressure at $ F $ is $ H - h + N $.

We must carefully distinguish this pressure on any square inch of the pipe, from the obstruction or resistance which it actually exerts, and which is part of the cause of the reservoir. The pressure, by the laws of hydrostatics, is the same with that exerted on the water by a square inch of piston or forcing head of water. This must balance the 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 the resistance which may not be balanced by the tendency while the water has to slide along it, provided the pipe be long enough; or if the pipe is too short, the tendency on the pipe may more than balance all the resistances obtained 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 the concentric films glide within those around them, somewhat like the sliding tubes of a spy-glass when we draw it out by taking hold of the end of the innermost. The second film passes beyond the first and 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 cannot be determined by theory. We see however that pipes of greater diameter must require a greater length, and in a proportion which is probably that of the number of filaments, or the square of the diameter. Du Buat found his 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 twenty-five feet. He therefore saw that the square of the diameter in inches, multiplied by seventy-two, 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 resistance occasion to be exerted there before they can be overcome. The resistance may be represented by $ \frac{d}{s} $, where $ 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 $ bd $. This multiplied by the length is $ bds $; and this multiplied by the slope $ \frac{1}{s} $ is $ bd $, the relative weight of the column whose length is $ s $. The relative weight of one inch is therefore $ \frac{bd}{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. The 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} \times 100 \times 62\frac{1}{2} = 6250 $ pounds nearly. Therefore an inch of one seam is strained by 2601 pounds. A rod of lead one inch square is pulled asunder by 860 pounds (see Strength of Materials). Therefore, if the thickness of the seam is $ \frac{860}{6250} $ 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 twelve inches diameter and sixty feet deep should be half an inch thick. But these things will be more properly computed by means of the list given in 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 greater 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 already determined, 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 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 eighteenth 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 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 inefficient 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 $2G$, and the section of the main be $A$, the supply at the entry of the main would be $A\sqrt{2GH} - x$; 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 $\sqrt{H} : \sqrt{x} = D : D\sqrt{\frac{x}{H}} = \delta$. 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, $y : H - y = x(H - y)$; and this fourth 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, $A\sqrt{2G}\sqrt{\frac{x(H - y)}{y}} = A\sqrt{2G}\sqrt{\frac{x(H - y)}{y}} + D\sqrt{\frac{x}{H}}$. From this we deduce the value of $x = \frac{2GHA^2}{(A\sqrt{2G}\sqrt{\frac{H - y}{y}} + D\sqrt{\frac{x}{H}})^2 + 2GA^2}$. This value of $x$ being substituted in the equation of the discharge $\delta$ 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 affects the train of the main, under the same head, by altering the discharge at its extremity, and the velocity in it between the branch; and if it be near the reservoir, it greatly affects the train, because the diminished velocity takes place through a greater extent, and there is a greater diminution of its resistances.

When the branch is taken off at a considerable distance from the reservoir, the problem becomes more complicated, and the head 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 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.

There be an inclined tube, having a head greater than what is competent to the velocity, we may bring it into train by 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, for the sake of economy, being generally of small dimensions, the velocity is much diminished. But, at the same time, it approaches near to that which the same conduit would bring directly from the reservoir, because its small velocity will produce a slight change in the train of the main conduit.

We should now treat of jets of water, which still make ornament in the magnificent pleasure-grounds of the wealthy. Some of these are indeed grand objects, such as those at Peterhoff in Russia, which spout about sixty feet high a column of nine inches diameter, that falls away, and shakes the ground with its blow. Even a spout of an inch or two inches diameter, lancing to the height of fifteen 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 trifling 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.

CLASS 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 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 book 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 Encyclopédie Méthodique, which particularly relate to the mechanical arts, 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, be compared together, the arches DB and EF are equal. The mechanical energy of the water contained in the bucket BD, 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 : bd = CF : CT, and EF × CT = bd × CF. Therefore, if the prism of water whose vertical section is bbdd, 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 ahka hung on at F. And the effect of any portion BF of this ring is the same with that of the corresponding portion bFfb of the vertical pillar. We do not take into account the small difference which arises from the depth B3 or Ff, because we may suppose the circle described through the centres of gravity of the buckets. And in the further 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 δ, and the bucket will only retain the quantity ZBD δ; and if the buckets are formed by partitions directed to the axis of the wheel, the whole water must be run out by the time that they descend to the level of the axis. To prevent this, many contrivances have been adopted. The wheel has been surrounded with a hoop or sweep, consisting of a circular board, which comes almost into contact with the rim of the wheel, and terminates at H, where the water is allowed to run off. But unless the work is executed with uncommon accuracy, the wheel made exactly round, and the sweep exactly fitting it, a great quantity of water escapes between them; and there is a very sensible obstruction to the motion of such a wheel, from something like friction between the water and the sweep. Frost also effectually stops the motion of such a wheel. Sweeps have therefore been generally laid aside, although there are situations where they might be used with good effect.

Millwrights have turned their whole attention to the giving a form to the buckets which shall enable them to retain the water along a great portion of the circumference of the wheel. It would be endless to describe all these contrivances; and we shall therefore content ourselves with one or two of the most approved. The intelligent reader will readily see that many of the circumstances which occur in producing the ultimate effect (such as the facility with which the water is received into the buckets, the place which it is to occupy during the progress of the bucket from the top to the bottom of the wheel, the readiness with which they are evacuated, or the chance that the water has of being dragged beyond the bottom of the wheel by its adhesion, &c. &c.) are such as do not admit of precise calculation on reasoning about their merits; and that this or that form can seldom be evidently demonstrated to be the very best possible. But, at the same time, he will see the general reasons of preference, and his attention will be directed to circumstances which must be attended to in order to have a good bucketed wheel.

Fig. 6 is the outline of a wheel having forty buckets. The ring of board contained between the concentric circles QDS and PAR, making the ends of the buckets, is called the shrouding, in the language of the art, and QP is called the depth of shrouding. The inner circle PAR is called the sole of the wheel, and usually consists of boards nailed to strong wooden rings of compass-timber of considerable scantling, firmly united with the arms or radii. The partitions which determine the form of the buckets consist of three different planes or boards AB, BC, CD, which are variously named by different artists. We have heard them named the start or shoulder, the arm, and the wrest (probably for wrist, on account of a resemblance of the whole line to the human arm); B is also called the elbow. Fig. 7 represents a small portion of the same bucketing on a larger scale, that the proportions of the parts may be more distinctly seen. AG, the sole of one bucket, is made about 1/4th more than the depth GH of the shrouding. The start AB is 1/2 of AI. The plane BC is so inclined to AB that it would pass through H; but it is made to terminate in C, in such a manner that FC is 1/8th of GH or AI. Then CD is so placed that HD is about 1/4th of IH.

By this construction, it follows that the area FABC is very nearly equal to DABC; so that the water which will fill the space FABC will all be contained in the bucket when it shall come into such a position that AD is a horizontal line; and the line AB will then make an angle of nearly 35° with the vertical, or the bucket will be 35° from the perpendicular. If the bucket descend so much lower that one half of the water runs out, the line AB will make an angle of 25°, or 24° nearly, with the vertical. Therefore the wheel, filled to the degree now mentioned, will begin to lose water at about 1/4th of the diameter from the bottom, and half of the water will be discharged from the lowest bucket, about 1/4th of the diameter farther down. These situations of the discharging bucket are marked at T and V in fig. 6. Had a greater proportion of the buckets been filled with water when they were under the spout, the discharge would have begun at a greater height from the bottom, and we should lose a greater portion of the whole fall of water. The loss by the present construction is less than 1/10th (supposing the water to be delivered into the wheel at the very top), and may be estimated at about 1/4th; for the loss is the versed sine of the angle which the radius of the bucket makes with the vertical. The versed sine of 35° is nearly 1/4th of the radius (being 0.18085), or 1/4th of the diameter. It is evident that if only 1/2 of this water were supplied to each bucket as it passes the spout, it would have been retained for 10° more of a revolution, and the loss of fall would have been only about 1/4th.

These observations serve to show, in general, that an advantage is gained by having the buckets so capacious that the quantity of water which each can receive as it passes the spout may not nearly fill it. This may be accomplished by making them of a sufficient length, that is, by making the wheel sufficiently broad between the two shroudings. Economy is the only objection to this practice, and it is generally very ill placed. When the work to be performed by the wheel is great, the addition of power gained by a greater breadth will soon compensate for the additional expense.

The third plane CD is not very frequent, and millwrights generally content themselves with continuing the board all the way from the elbow B to the outer edge of the wheel at H; and AB is generally no more than one third of the depth AI. But CD is a very evident improvement, causing the wheel to retain a very sensible addition to the water. Some indeed make this addition more considerable, by bringing BC more outward, so as to meet the rim of the wheel at H, for instance, and making HD coincide with the rim. But this makes the entry of the water somewhat more difficult during the very short time that the opening of the bucket passes the spout. To facilitate this as much as possible, the water should get a direction from the spout such as will send it into the buckets in the most perfect manner. This may be obtained by delivering water through an aperture that is divided by thin plates of board or metal, placed in the proper position, as we have represented in Fig. 6. The form of bucket last mentioned, having the wrest concentric with the rim, is unfavourable to the ready adhesion of the water; whereas an oblique wrest conducts the water which has missed one bucket into the next below.

The mechanical consideration of this subject also shows us that a deep shrouding, in order to make a capacious bucket, is not a good method: it does not make the buckets retain their water any longer, and it diminishes the effective fall of water; for the water received at the top of the wheel immediately falls to the bottom of the bucket, and thus shortens the fictitious pillar of water, which we showed to be the measure of the effective or useful pressure on the wheel; and this concurs with our former reasons for recommending as great a breadth of the wheel, and length of buckets, as economical considerations will permit.

bucket-wheel was some

Fig. 8.

improvement executed by Mr Robert Bells, at the cotton-mills of Holton, of a construction entirely new, but founded on a good principle, which is susceptible of great extension. It is represented in fig. 8. The bucket consists of a start AB, an arm BC, and a wrest CD, concentric with the rim. But the bucket is also divided by a partition LM, concentric with the sole and rim, and so placed as to make the inner and outer portions of nearly equal capacity. It is evident, without any further reasoning about it, that this partition will enable the bucket to retain its water much longer. When they are filled one third, they retain the whole water at 18° from the bottom, and they retain half at 11°. They do not admit the water quite so freely as buckets of the common construction; but by means of the contrivance mentioned a little ago for the sea (also the invention of Mr Burns, and furnished with air-work, which raised or depressed it as the supply of water varied, so as at all times to employ the whole fall of the water), it is found that a slow moving wheel allows one half of the water to get into the inner buckets, especially if the partition do not altogether reach the radius drawn through the lip D of the outer bucket.

This is a very great improvement of the bucket-wheel; and when the wheel is made of a liberal breadth, so that the water may be very shallow in the buckets, it seems to affect the performance as far as it can go. Mr Burns made the first trial on a wheel of twenty-four feet diameter; and its performance is manifestly superior to that of the wheel when it replaced, and which was a very good one. It has another valuable property: when the supply of water is very scanty, a proper adjustment of the apparatus in the sea will direct almost the whole of the water into the inner buckets, which, by placing it at a greater distance from the axis, makes a very sensible addition to its mechanical energy.

It is said that this principle is susceptible of considerable extension; and it is evident that two partitions will increase the effect, and that it will increase with the number of partitions; so that when the practice now begun, of making water-wheels of iron, shall become general, and therefore very thin partitions are used, their number may be greatly increased without any inconvenience, and it is obvious that the series of partitions must greatly contribute to the stiffness and general firmness of the whole wheel.

There frequently occurs a difficulty in the making of bucket-wheels, when the half-taught mill-wright attempts to retain the water a long time in the buckets. The water gets into them with a difficulty which he cannot account for, and spills all about, even when the buckets are not moving away from the spout. This arises from the air, which must find its way out to admit the water, but is obstructed by the entering water, and occasions a great spluttering at the entry. This may be entirely prevented by making the spout considerably narrower than the wheel, which will leave room at the two ends of the buckets for the escape of the air. This obstruction is vastly greater than one would imagine, for the water drags along with it a great quantity of air, as is evident in the water-blast described by many authors.

There is another and very serious obstruction to the motion of an overshot or bucketed wheel. When it moves in back-water, it is not only resisted by the water when it moves more slowly than the wheel, which is very frequently the case, but it lifts a great deal in the rising buckets. In some particular states of back-water, the descending bucket fills itself completely with water, and in other cases it contains a very considerable quantity, and air of common density; while in some rarer cases it contains less water, with air in a condensed state. In the first case the rising bucket must come up filled with water, which it cannot drop till its mouth get out of the water. In the second case, part of the water goes out before this, but the air rarefies, and therefore there is still some water dragged or lifted up by the wheel, by suction, as it is usually called. In the last case there is no such back-load on the rising side of the wheel, but (which is as detrimental to its performance) the descending side is employed in condensing air; and although this air aids the ascent of the rising side, it does not aid it so much as it impedes the descending side, being (by the form of the bucket) nearer to the vertical line drawn through the axis.

All this may be completely prevented by a few holes made in the start of each bucket. Air being at least 800 times rarer than water, will escape through a hole almost thirty times faster with the same pressure. Very moderate holes will therefore suffice for this purpose; and the small quantity of water which these holes discharge during the descent of the buckets, produces a loss which is altogether insignificant. The water which runs out of one runs into another, so that there is only the loss of one bucket. We have seen a wheel of only fourteen feet diameter working in nearly three feet of back-water. It laboured prodigiously, and brought up a great load of water, which fell from it in abrupt dashes, rendering the motion very bobbling. When three holes of an inch diameter were made in each bucket (twelve feet long), the wheel laboured no more, there was no more plunging of water from its rising side, and its power on the machinery was increased more than one fourth.

These practical observations may contain information that is new even to several experienced millwrights. To persons less informed they cannot fail of being useful. We now proceed to consider the action of water thus lying in the buckets of a wheel, and to ascertain its energy as it may be modified by different circumstances of fall, velocity, &c.

With respect to variations in the fall, there can be little room for discussion. Since the active pressure is measured by the pillar of water reaching from the horizontal plane where it is delivered on the wheel, to the horizontal plane where it is spilled by the wheel, it is evident that it must be proportional to this pillar, and therefore we must deliver it as high and retain it as long as possible.

This maxim obliges us, in the first place, to use a wheel whose diameter is equal to the whole fall. We shall not gain anything by employing a larger wheel; for although we should gain by using only that part of the circumference where the weight will act more perpendicularly to the radius, we shall lose more by the necessity of discharging the water at a greater height from the bottom: for we must suppose the buckets of both the wheels equally well constructed; in which case, the heights above the bottom where they will discharge the water, will increase in the proportion of the diameter of the wheel. Now, that we shall lose more by this than we gain by the more direct application of the weight, is plain, without any further reasoning, by taking the extreme case, and supposing our wheel enlarged to such a size that the useless part below is equal to the whole fall. In this case the water will be spilled from the buckets as soon as it is delivered into them. All intermediate cases therefore partake of the imperfection of this.

When the fall is exceedingly great, a wheel of an equal diameter becomes enormously large and expensive, and is of itself an unmanageable load. We have however seen wheels of fifty-eight feet diameter, which worked extremely well; but they are of very difficult construction, and extremely apt to warp and go out of shape by their weight. In cases like this, where we are unwilling to lose any part of the force of a small stream, the best form of a bucket-wheel is an inverted chain-pump. Instead of employing a chain-pump of the best construction, ABCDEA (fig. 9), to raise water through the upright pipe CB, by means of a force applied to the upper wheel A, let the water be delivered from a spout F, into the upper part of a pipe BC, and it will press down the plugs in the lower and narrower bored part of it with the full weight of the column, and escape at the dead level of C. This weight will urge round the wheel A without any defalcation; and this is the most powerful manner in which any fall of water whatever can be applied, and it exceeds the most perfect overshot wheel. But though it excels all chains of buckets in economy and in effect, it has all the other imperfections of this kind of machinery. Though the chain of plugs be of great strength, it has so much motion in its joints that it needs frequent repairs; and when it breaks, it is generally in the neighbourhood of A, on the loaded side, and all comes down with a great crash. There is also a loss of power by the immersion of so many plugs and chains in the water; for there can be no doubt but that if the plugs were large enough and light enough, they would buoy and even draw up the plug in the narrow part at C. They must therefore diminish, in all other cases, the force with which this plug is pressed down.

The velocity of an overshot wheel is a matter of very great nicety; and authors, both speculative and practical, have entertained different, nay opposite, opinions on the subject. M. Belidor, whom the engineers of Europe have long been accustomed to regard as sacred authority, maintains that there is a certain velocity related to that obtainable by the whole fall, which will procure to an overshot wheel the greatest performance. Desaguliers, Smeaton, Lambert, De Parcieux, and others, maintain that there is no such relation, and that the performance of an overshot wheel will be the greater, as it moves more slowly by an increase of its load of work. Belidor maintains that the active power of water lying in a bucket-wheel of any diameter is equal to that of the impulse of the same water on the floats of an undershot wheel when the water issues from a sluice in the bottom of the dam. The other writers whom we have named assert that the energy of an undershot wheel is but one half of that of an overshot, actuated by the same quantity of water falling from the same height.

To a manufacturing country like ours, that derives astonishing superiority, by which it more than compensates for the impediments of heavy taxes and luxurious living, chiefly from its machinery, in which it leaves all Europe far behind, the decision of this question, in such a manner as shall leave no doubt or misconception in the mind even of an unlettered artist, must be considered as a material service, and we think that this is easily attainable.

When any machine moves uniformly, the accelerating force or pressure actually exerted on the impelled point of the machine is in equilibrium with all the resistances which are exerted at the working point, with those arising from friction, and those that are excited in different parts of the machine by their mutual actions. This is an incontestable truth, and though little attended to by the mechanicians, is the foundation of all practical knowledge of machines. Therefore, when an overshot wheel moves uniformly, with any velocity whatever, the water is acting with its whole weight, for gravity would accelerate its descent, if not completely balanced by some re-action; and in this balance gravity and the re-acting part of the machine exert equal and opposite pressures, and thus produce the uniform motion of the machine. We are thus particular on this point, because we observe mechanicians of the first name employing a mode of reasoning on the question now before us which is specious, and appears to prove the conclusion which they draw, but is nevertheless contrary to true mechanical principles. They assert that the slower a heavy body is descending (suppose in a scale suspended from an axis in peritrochoea), the more does it press on the scale, and the more does it urge the machine round; and therefore the slower an overshot wheel turns, the greater is the force with which the water urges it round, and the more work will be done. It is very true that the machine is more forcibly impelled, and that more work is done; but this is not because a pound of water presses more strongly, but because there is more water pressing on the wheel, for the spent supplies at the same rate, and each bucket receives more water as it passes by it.

Let us therefore examine this point by the unquestionable principles of mechanics.

Let the overshot wheel AFH (fig. 5) receive the water from a spout at the very top of the wheel; and in order that the wheel may not be retarded by dragging into motion the water simply laid into the uppermost bucket at A, let it be received at B, with the velocity (directed in a tangent to the wheel) acquired by the head of water AB. This velocity therefore must be equal to that of the rim of the wheel. Let this be v, or let the wheel and the water move over v inches in a second. Let the buckets be of such dimensions, that all the water which each receives as it passes the spout is retained till it comes to the position R, where it is discharged at once. It is plain that, in place of the separate quantities of water lying in each bucket, we may substitute a continued ring of water, equal to their sum, and uniformly distributed in the space BERF. This constitutes a ring of uniform thickness. Let the area of its cross section EB and EF be called a. We have already demonstrated, that the mechanical energy with which this water on the circumference of the wheel urges it round is the same with what would be exerted by the pillar bwe pressing on Ff, or acting by the lever CF. The weight of this pillar may be expressed by $a \times br$, or $a \times PS$; and if we call the radius CF of the wheel R, the momentum or mechanical energy of this weight will be represented by $a \times PS \times R$.

Now, let us suppose that this wheel is employed to raise a weight W, which is suspended by a rope wound round the axis of the wheel. Let r be the radius of this axle. Then $W \times r$ is the momentum of the work. Let the weight rise with the velocity u when the rim of the wheel turns with the velocity v, that is, let it rise u inches in a second. Since a perfect equilibrium obtains between the power and the work when the motion is uniform, we must have $W \times r = a \times PS \times R$. But it is evident that $R = r + v$. Therefore $W \times u = a \times v \times PS$.

Now the performance of the machine is undoubtedly measured by the weight and the height to which it is raised in a second, or by $W \times u$. Therefore the machine is in its be possible state when $a \times v \times PS$ is a maximum. But it is plain that $a \times v$ is an invariable quantity, for it is the cubic inches of water which the spout supplies in a second.

The wheel moves fast, little water lies in each bucket, and $v$ is small. When $v$ is small, $a$ is great, for the opposite reason; but $a \times v$ remains the same. Therefore we must make $PS$ a maximum, that is, we must deliver the water as high as possible. But this diminishes $AP$, and this diminishes the velocity of the wheel; and as this has no limit, this proposition is demonstrated, and an overshot wheel does more work as it moves slowest.

Convincing as this discussion must be to any mechanic, we are anxious to impress the same maxim on the minds of practical men, unaccustomed to mathematical reasoning of any kind. We therefore beg indulgence for doing a popular view of the question, which requires no investigation.

We may reason in this way. Suppose a wheel having thirty buckets, and that six cubic feet of water are delivered in a second on the top of a wheel, and discharged without any loss by the way at a certain height from the bottom of the wheel. Let this be the case whatever is the rate of the wheel's motion, the buckets being of a sufficient capacity to hold all the water which falls into them. Let this wheel be employed to raise a weight of any kind, suppose water in a chain of thirty buckets, to the same height as with the same velocity. Suppose further, that when loaded on the rising side of the machine is one half of that same wheel, the wheel makes four turns in a minute, or ten turns in fifteen seconds. During this time ninety cubic feet of water have flowed into the thirty buckets, and each has received three cubic feet. Then each of the rising buckets contains $1\frac{1}{2}$ feet, and forty-five cubic feet are delivered into the upper cistern during one turn of the wheel, or 180 cubic feet in one minute.

Now, suppose the machine so loaded, by making the rising buckets more capacious, that it makes only two turns in a minute, or one turn in thirty seconds. Then each descending bucket must contain six cubic feet of water. If each bucket of the rising side contained three cubic feet, the motion of the machine would be the same as before. This is a point which no mechanician will controvert. When two pounds are suspended to one end of a string which passes over the pulley, and one pound to the other end, the descent of the two pounds will be the same with that of a four-pound weight which is employed in the same manner to draw up two pounds. Our machine would therefore continue to make four turns in the minute, and would deliver ninety cubic feet during each turn, and 360 in a minute.

By supposition, it is making but two turns in a minute; it must proceed from a greater load than three cubic feet of water in each rising bucket. The machine must therefore be raising more than ninety feet of water during one turn of the wheel, and more than 180 in the minute.

Thus it appears, that if the machine be turning twice as fast as before, there is more than twice the former quantity of rising buckets, and more will be raised in a minute by the same expenditure of power. In like manner, if the machine go three times as slow, there must be more than three times the former quantity of water in the rising buckets, and more work will be done.

But we may go further, and assert, that the more we revolve the machine, by loading it with more work of a similar kind, the greater will be its performance. This does immediately appear from the present discussion: but let us call the first quantity of water in the rising bucket the water raised by four turns in a minute will be $4 \times 30 \times A = 120A$. The quantity in this bucket, when the machine goes twice as slow, has been shown to be greater than $2A$ (call it $2A + x$); the water raised by two turns in a minute will be $2 \times 30 \times (2A + x) = 120A + 60x$. Now, let the machine go four times as slow, making but one turn in a minute, the rising bucket must now contain more than twice $2A + x$, or more than $4A + 2x$; call it $4A + 2x + y$. The work done by one turn in a minute will now be $30 \times (4A + 2x + y) = 120A + 60x + 30y$.

By such an induction of the work, done with any rates of motion we choose, it is evident that the performance of the machine increases with every diminution of its velocity that is produced by the mere addition of a similar load of work, or that it does the more work the slower it goes.

We have supposed the machine to be in its state of permanent uniform motion. If we consider it only in the beginning of its motion, the result is still more in favour of slow motion: for, at the first action of the moving power, the inertia of the machine itself consumes part of it, and it acquires its permanent speed by degrees, during which the resistances arising from the work, friction, &c., increase, till they exactly balance the pressure of the water; and after this the machine accelerates no more. Now the greater the power and the resistance arising from the work are in proportion to the inertia of the machine, the sooner will all arrive at its state of permanent velocity.

There is another circumstance which impairs the performance of an overshot wheel moving with a great velocity, viz. the effects of the centrifugal force on the water in the buckets. Our millwrights know well enough, that too great velocity will throw the water out of the buckets; but few, if any, know exactly the diminution of power produced by this cause. The following very simple construction will determine this. Let $AOB$ (fig. 10) be an overshot wheel, of which $AB$ is the upright diameter, and $C$ is the centre. Make $CF$ the length of a pendulum which will make two vibrations during one turn of the wheel. Draw $FE$ to the elbow of any of the buckets. The water in this bucket, instead of having its surface horizontal, as $NO$, will have it in the direction $NO$ perpendicular to $FE$ very nearly.

For the time of falling along half of $FC$ is to that of two vibrations of this pendulum, or to the time of a revolution of the wheel, as the radius of a circle is to its circumference; and it is well known that the time of moving along half of $AC$, by the uniform action of the centrifugal force, is to that of a revolution as the radius of a circle to its circumference. Therefore the time of describing one half of $AC$ by the centrifugal force, is equal to the time of describing one half of $FC$ by gravity. These spaces, being similarly described in equal times, are proportional to the accelerating forces. Therefore $\frac{1}{2} FC : \frac{1}{2} AC$, or $FC : AC =$ gravity : centrifugal force. Complete the parallelogram $FCEK$. A particle at $E$ is urged by its weight in the direction $KE$ with a force which may be expressed by $FC$ or $KE$; and it is urged by the centrifugal force in the direction $CE$, with a force $= AC$ or $CE$. By their combined action it is urged in the direction $FE$. Therefore, as the surface of standing water is always at right angles to the action of gravity, that is, to the plumb-line, so the surface of the water in the revolving bucket is perpendicular to the action of the combined force $FE$.

Let $NEO$ be the position of the bucket, which just holds all the water which it received as it passed the spout when not affected by the centrifugal force; and let $NDO$ be its position when it would be empty. Let the vertical lines through $D$ and $E$ cut the circle described round $C$ with the radius CF in the points H and I. Draw HC, IC, cutting the circle AOB in L and M. Make the arch d' equal to AL, and the arch e' equal to AM: then C and C will be the positions of the bucket on the revolving wheel, corresponding to CD and CEO on the wheel at rest. Water will begin to run out at s, and it will be all gone at d. The demonstration is evident.

The force which now urges the wheel is still the weight really in the buckets; for though the water be urged in the direction with the force FE, one of its constituents, CE, has no tendency to impel the wheel; and KE is the only impelling force.

It is but of late years that mills have been constructed or attended to with that accuracy and scientific skill which are necessary for deducing confident conclusions from any experiments that can be made with them; and it is therefore no matter of wonder that the opinions of millwrights have been so different on this subject. There is a natural wish to see a machine moving briskly; it has the appearance of activity; but a very slow motion always looks as if the machine were overloaded. For this reason millwrights have always yielded slowly, and with some reluctance, to the repeated advices of the mathematicians: but they have yielded, and we see them adopting maxims of construction more agreeable to sound theory, making their wheels of great breadth, and loading them with a great deal of work. Euler says that the performance of the best mill cannot exceed that of the worst above one fifth; but we have seen a stream of water completely expended in driving a small flax-mill, which now drives a cotton-mill of 4000 spindles, with all its carding, roving, and drawing machinery, besides the lathes and other engines of the smith and carpenter's workshops, exerting a force not less than ten times what sufficed for the flax-mill.

The above discussion only demonstrates in general the advantage of slow motion, but does not point out in any degree the relation between the rate of motion and the work performed, nor even the principles on which it depends. Yet this is a subject fit for a mathematical investigation, and we would prosecute it in this place if it were necessary for the improvement of practical mechanics. But we have seen that there is not, in the nature of things, a maximum of performance attached to any particular rate of motion which should therefore be preferred. For this reason we omit this discussion of mere speculative curiosity. It is very intricate; for we must now express the pressure on the wheel by a constant pillar of water incumbent on the extremity of the horizontal arm, as we did before when we supposed the buckets completely filled; nor by a smaller constant pillar, corresponding to a smaller but equal quantity lying in every bucket. Each different velocity puts a different quantity of water into the bucket as it passes the spout, and this occasions a difference in the place where the discharge is begun and completed. This circumstance is some obstacle to the advantages of very slow motions, because it brings on the discharge sooner.

All this may indeed be expressed by a simple equation of easy management; but the whole process of the mechanical discussion is both intricate and tedious, and the results are so much diversified by the forms of the buckets, that they do not afford any rule of sufficient generality to reward our trouble. The curious reader may see a very full investigation of this subject in two dissertations by Elvius in the Swedish Transactions, and in the Hydrodynamique of Professor Kästner of Göttingen, who has abridged these dissertations of Elvius, and considerably improved the whole investigation, and has added some comparisons of his deductions with the actual performance of some great works. These comparisons however are not very satisfactory. There is also a very valuable paper on this subject by Lambert, in the Memoirs of the Academy of Berlin for the year 1775. From these dissertations, and from the Hydrodynamique of the Abbé Bossut, the reader will acquire all that theory can teach of the relation between the pressures of the power and work on the machine, and the rates of its motion. The practical reader may rest with confidence on the simple demonstration which we have given, that the performance is improved by diminishing the velocity.

All we have to do, therefore, is to load the machine, and thus to diminish its speed, unless other physical circumstances throw obstacles in the way; but there are such obstacles. In all machines there are little inequalities of action that are unavoidable. In the action of a wheel and pinion, though made with the utmost judgment and care, there are such inequalities. These increase by the changes of form occasioned by the wearing of the machine; much greater irregularities arise from the subsultory motions of cranks, stampers, and other parts which move unequally or reciprocally. A machine may be so loaded as just to be in equilibrio with its work, in the favourable position of its parts. When this changes into one less favourable, the machine may stop; if not, it at least staggers, hobbles, or works unequally. The rubbing parts bear long on each other, with enormous pressures, and cut deep, and increase friction. Such slow motions must therefore be avoided. A little more velocity enables the machine to get over those increased resistances by its inertia, or the great quantity of motion inherent in it. Great machines possess this advantage in a superior degree, and will therefore work steadily with a smaller velocity. These circumstances are hardly susceptible of mathematical discussion, and our best reliance is on well-directed experience.

For this purpose, the reader will do well to peruse with care the excellent paper by Mr Smeaton in the Philosophical Transactions for 1759. This dissertation contains a numerous list of experiments, most judiciously contrived by him, and executed with the accuracy and attention to the most important circumstances which is to be observed in all that gentleman's performances.

It is true, these experiments were made with small models; and we must not, without great caution, transfer the results of such experiments to large works. But we may safely transfer the laws of variation which result from a variation of circumstances, although we must not adopt the absolute quantities of the variations themselves. Mr Smeaton was fully aware of the limitations to which conclusions drawn from experiments on models are subject, and has made the applications with his usual sagacity.

His general inference is, that, in smaller works, the rim of the overshot wheel should not have a greater velocity than three feet in a second; but that larger mills may be allowed a greater velocity than this. When every thing is executed in the best manner, he says that the work performed will amount to fully two thirds of the power expended; that is, that three cubic feet of water descending from any height will raise two to the same height.

It is not very easy to compare these deductions with observations on large works; because there are few cases where we have good measures of the resistances opposed by the work performed by the machine. Mills employed for pumping water afford the best opportunities. But the inertia of their working gear diminishes their useful performance very sensibly, because their great beams, pump-rods, &c. have a reciprocating motion, which must be destroyed and produced anew in every stroke. We have examined some machines of this kind which are esteemed good ones, and we find few of them whose performance exceeds one half of the power expended.

By comparing other mills with these, we obtain the best information of their resistances. The comparison with mills worked by Watt and Boulton's steam-engines, is per- have better measure of the resistances opposed by different kinds of work, because their power is very distinctly seen. We have been informed by one of the most eminent engineers, that a ton and a half of water per minute will grind and dress one bushel of wheat per hour. This is equivalent to nine tons falling ten feet.

If an overshot wheel opposed no resistance, and only one scket were filled, the wheel would acquire the velocity of a fall through the whole height. But when it is in its state of accelerated motion, if another bucket of water delivered into it, its motion must be checked at the first, by the necessity of dragging forward this water. If buckets fill in succession as they pass the spout, the velocity acquired by an unresisting wheel is but half of that which one bucket would give. In all cases, therefore, velocity is diminished by the inertia of the entering water when it is simply laid into the upper buckets. The performance will therefore be improved by delivering the water on the wheel with that velocity with which the wheel is really moving. And as we cannot give the direction of a tangent to the wheel, the velocity with which it is delivered on the wheel must be so much greater than the needed velocity of the rim, that it shall be precisely equal to it when it is estimated in the direction of the tangent. Three or four inches of fall are sufficient for this purpose; and it should never be neglected, for it has a very sensible influence on the performance. But it is highly improper to give it more than this, with the view of impelling the wheel by its stroke; for even although it were proper to employ part of the fall in this way (which we shall presently see to be very improper), we cannot procure his impulse; because the water falls among other things it strikes the boards of the wheel with such obstinacy that it cannot produce any such effect.

It is a much debated question among millwrights, whether the diameter of the wheel should be such that the water will be delivered at the top of the wheel; or so large that the water is received at some distance from the top, where it will act more perpendicularly to the arm. We apprehend that the observations formerly made will decide in favour of the first practice. The space below, where the water is discharged from the wheel, being proportional to the diameter of the wheel, there is an undoubted advantage attending a large wheel; and this is not compensated by delivering the water at a greater distance from the perpendicular. We should therefore recommend the use of the whole descending side, and make the diameter of the wheel no greater than the fall, till it is so much reduced that the centrifugal force begins to produce a sensible effect. Since the rim can hardly have a smaller velocity than three feet per second, it is evident that a small wheel must revolve more rapidly. This made it proper to describe the determination that we have given of the loss of power produced by the centrifugal force. But even with this view, we should employ much smaller wheels than are generally done on small falls. Indeed the loss of water at the bottom may be diminished, by nicely fitting the arch which surrounds the wheel, so as not to allow the water to escape by the sides or bottom. While this improvement remains in good order, and the wheel entire, it produces a very sensible effect; but the passage widens continually by the wearing of the wheel. A piece of wood or stone falling about the wheel tears off part of the shrouding or rock, and frosty weather frequently binds all fast. It rarely seldom answers expectations. We have nothing to say on this case to what we have already extracted from Mr. Seaton's Dissertation on the subject of Breast or Half Breast Wheels.

Tire is another form of wheel by which water is made to act on a machine by its weight, which merits consideration. It is known in this country by the name of Barker's mill, and has been described by Desaguliers, vol. ii. p. 460. It consists of an upright pipe or trunk AB (fig. 11), communicating with two horizontal branches BC, Be, which have a hole Ce near their ends, opening in opposite directions, at right angles to their lengths. Suppose water to be poured in at the top from the spout F, it will run out by the holes C and e with the velocity corresponding to the depth of these holes under the surface. The consequence of this must be, that the arms will be pressed backwards; for there is no solid surface at the hole C, on which the lateral pressure of the water can be exerted, while it acts with its full force on the opposite side of the arm. This unbalanced pressure is equal to the weight of a column having the orifice for its base, and twice the depth under the surface of the water in the trunk for its height. This measure of the height may seem odd, because if the orifice were shut, the pressure on it is the weight of a column reaching from the surface. But when it is open, the water issues with nearly the velocity acquired by falling from the surface, and the quantity of motion produced is that of a column of twice this length, moving with this velocity. This is actually produced by the pressure of the fluid, and must therefore be accompanied by an equal reaction.

Now suppose this apparatus set on the pivot E, and to have a spindle AD above the trunk, furnished with a cylindrical bobbin D, having a rope wound round it, and passing over a pulley G. A weight W may be suspended there, which may balance this backward pressure. If the weight be too small for this purpose, the retrograde motion of the arms will wind up the cord, and raise the weight; and thus we obtain an acting machine, employing the pressure of the water, and applicable to any purpose. A runner millstone may be put on the top of the spindle; and we should then produce a flour-mill of the greatest simplicity, having neither wheel nor pinion, and subject to hardly any wear. It is somewhat surprising, that although this was invented at the beginning of last century, and appears to have such advantage in point of simplicity, it has not come into use. So little has Dr Desaguliers's account been attended to (although it is mentioned by him as an excellent machine, and as highly instructive to the hydraulist), that the same invention was again brought forward by a German professor (Segner) as his own, and has been honoured by a series of elaborate disquisitions concerning its theory and performance by Euler and by John Bernoulli. Euler's Dissertations are to be found in the Memoirs of the Academy of Berlin, 1751, &c. and in the Nov. Comment. Petropol. tom. vi. Bernoulli's are at the end of his Hydraulics. Both these authors agree in saying that this machine excels all other methods of employing the force of water. Simple as it appears, its true theory, and the best form of construction, are most abstruse and delicate subjects; and it is not easy to give such an account of its principles as will be understood by an ordinary reader.

We see in general that the machine must press backwards; and little investigation suffices for understanding the intensity of this pressure when the machine is at rest. But when it is allowed to run backwards, withdrawing itself from the pressure, the intensity of it is diminished; and if no other circumstances intervened, it might not be difficult to say what particular pressure corresponded to any rate of motion. Accordingly, Desaguliers, presuming on the simplicity of the machine, affirms the pressure to be the weight of a column which would produce a velocity of efflux equal to the difference of the velocity of the fluid and of the machine; and hence he deduces that its performance will be the greatest possible when its retrograde velocity is one third of the velocity acquired by falling from the surface, in which case it will raise $\frac{1}{3}$ths of the water expended to the same height, which is double of the performance of a mill acted on by the impulse of water.

But this is a very imperfect account of the operation. When the machine (constructed exactly as we have described) moves round, the water which issues descends in the vertical trunk, and then, moving along the horizontal arms, partakes of this circular motion. This excites a centrifugal force, which is exerted against the ends of the arms by the intervention of the fluid. The whole fluid is subjected to this pressure (increasing for every section across the arm in the proportion of its distance from the axis), and every particle is pressed with the accumulated centrifugal forces of all the sections that are nearer to the axis. Every section therefore sustains an actual pressure proportional to the square of its distance from the axis. This increases the velocity of efflux, and this increases the velocity of revolution; and this mutual co-operation would seem to terminate in an infinite velocity of both motions. But, on the other hand, this circular motion must be given anew to every particle of water as it enters the horizontal arm. This can be done only by the motion already in the arm, and at its expense. Thus there must be a velocity which cannot be overpassed even by an unloaded machine. But it is also plain, that by making the horizontal arm very capacious, the motion of the water from the axis to the jet may be made very slow, and much of this diminution of circular motion prevented. Accordingly, Euler has recommended a form by which this is done in the most eminent degree. His machine consists of a hollow conoidal ring, of which fig. 12 is a section. The part AHha is a sort of funnel-basin, which receives the water from the spout F, not in the direction pointing towards the axis, but in the direction, and with the precise velocity, of its motion. This prevents any retardation by dragging forward the water. The water then passes down between the outer conoid ACea and the inner conoid HGgh along spiral channels formed by partitions soldered to both conoids. The curves of these channels are determined by a theory which aims at the annihilation of all unnecessary and improper motions of the water, but which is too abstruse to find a place here. The water thus conducted arrives at the bottom CGeg. On the outer circumference of this bottom is arranged a number of spouts (one for each channel), which are all directed one way in tangents to the circumference.

Adopting the common theory of the re-action of fluids, this should be a very powerful machine, and should raise $\frac{1}{3}$ths of the water expended. But if we admit the reaction to be equal to the force of the issuing fluid (and we do not see how this can be refused), the machine must be nearly twice as powerful. We therefore repeat our wonder that it has not been brought into use. But it appears that no trial has been made even of a model; so that we have no experiments to encourage an engineer to repeat the trial. Even the late author Professor Segner has not related anything of this kind in his *Exercitationes Hydraulicae*, where he particularly describes the machine. Such remissness has probably proceeded from fixing the attention on Euler's improved construction. It is plain that this must be a most cumbrous mass, even in a small size requiring a prodigious vessel, and carrying an unwieldy load. If we examine the theory which recommends this construction, we find that the advantages, though real and sensible, bear but a small proportion to the whole performance of the simple machine as invented by Dr Barker. It is therefore to be regretted that engineers have not attempted to realize the first project. We beg leave to recommend it, with a further argument taken from an addition made to it by M. Mathon de la Cour, in Rozier's *Journal de Physique*, January and August 1775. This gentleman brings down a large pipe FEH (fig. 13) from a reservoir, bends it upward at H, and introduces it into two horizontal arms DA, DB, which have an upright spindle DK, carrying a millstone in the style of Dr Barker's mill. The ingenious mechanician will have no difficulty of contriving a method of joining these pipes, so as to permit a free circular motion without losing much water. The operation of the machine in this form is evident. The water, pressed by the column FG, flows out at the holes A and B, and the unbalanced pressure on the opposite sides of the arms forces them round. The compendiousness and other advantages of this construction are more striking, allowing us to make use of the greatest fall without any increase of the size of the machine. It undoubtedly enables us to employ a stream of water too scanty to be employed in any other form. The author gives the dimensions of an engine which he had seen at Bourg Argental. AB is 92 inches, and its diameter 3 inches; the diameter of each orifice is $1\frac{1}{2}$; FG is 21 feet; the pipe D was fitted into C by grinding, and the internal diameter of D is 2 inches.

When the machine was performing no work, or was unloaded, and emitted water by one hole only, it made 115 turns in a minute; thus giving a velocity of forty-six feet per second for the hole. This is a curious fact; for the water would issue from this hole at rest with the velocity of $37\frac{1}{2}$. This great velocity (which was much less than the velocity with which the water actually quitted the pipe) was undoubtedly produced by the prodigious centrifugal force, which was nearly seventeen times the weight of the water in the orifice.

The empty machine weighed eighty pounds, and its weight was half supported by the upper pressure of the water, so that the friction of the pivots was much diminished. It is a pity that the author has given no account of any work done by the machine. Indeed it was only working ventilators for a large hall. His theory by no means embraces all its principles, nor is it well founded.

We think that the free motion round the neck of the feeding pipe, without any loss of water or any considerable friction, may be obtained in the following manner. AB (fig. 14) represents a portion of the revolving horizontal pipe, and CEec part of the feeding pipe. The neck of the first is turned truly cylindrical, so as to turn easily, but without shake, in the collar Ce of the feeding pipe, and each has a shoulder which may support the other. That the friction of this joint may not be great, and the pipes destroy each other by wearing, the horizontal pipe has an iron spindle EF, fixed exactly in the axis of the joint, and resting with its pivot F in a step of hard steel, fixed to the iron bar GH, which goes across the feeding pipe, and is firmly supported in it. This pipe is made bell-shaped, widening below. A collar or hose of thin leather is fitted to the inside of this pipe, and is represented (in section) by LKMmhl. It is kept in its place by means of a metal or wooden ring Nn, thin at the upper edge, and taper-shaped. This is drawn in above the leather, and stretching it, causes it to apply to the side of the pipe all around. There can be no leakage at this joint, because the water will press the leather to the smooth metal pipe; nor can there be any sensible friction, because the water gets at the edge of the leather, and the whole unbalanced pressure...

is the small crevice between the two metal shoulders. The shoulders need not touch, so that the friction must be sensible. We imagine that this method of tightening a tenon joint may be used with great advantage in many cases.

We have only further to observe on this engine, that any impaction by which the passage of the water is diminished obstructed produces a saving of water, which is in exact proportion to the diminution of effect. The only inconvenience that is not thus compensated, is when the jets are not right angles to the arms.

We repeat our wishes, that engineers would endeavour to bring this machine into use, seeing many situations where it may be employed to great advantage. Suppose, for instance, a small supply of water from a great height applied in a manner to a centrifugal pump, or to a hair belt passing over a pulley, and dipping in the water of a deep well. This would be a hydraulic machine exceeding all others in simplicity and durability, though inferior in effect to some other constructions.

Of Undershot Wheels.

A wheel goes by this name where the motion of the water is quicker than that of the partitions or boards of the wheel, and therefore impels them. These are called the floatboards, or floats, of an undershot wheel. The water, running in a mill-row, with a velocity derived from a head of water or from a declivity of channel, strikes on these floats, and occasions, by its deflections sidewise and upwards, a pressure on the floats sufficient for impelling the wheel.

There are few points of practical mechanics that have been more considered than the action of water on the floats of a wheel; hardly a book of mechanics being silent on the subject. But the generality of them, at least such as are intelligible to persons who are not very much conversant in chemical and mathematical discussion, have hardly done anything more than copied the earliest deductions from the same theory of the resistance of fluids. The consequence has been, that our practical knowledge is very imperfect; and it is chiefly from experience that we must still learn the performance of undershot wheels. Unfortunately this stops the improvement; because those who have the only opportunities of making the experiments are not sufficiently acquainted with the principles of hydraulics, and are apt to make differences in their performance to trifling nostrums in their construction, or in the manner of applying the impulse of the water.

We have said so much on the imperfection of our theories of the impulse of fluids in the article Resistance of Fluids, that we need not here repeat the defects of the common conceptions of the motions of undershot wheels. The part of the theory of the impulse of fluids which agrees best with observation is, that the impulse is in the duplicate proportion of the velocity with which the water strikes the float; that is, if v be the velocity of the stream, and u the velocity of the float, we shall have F, the impulse on the float when highest, to its impulse f on the float moving with the velocity, as $v^2$ to $(v-u)^2$, and $f = F \times \frac{(v-u)^2}{v^2}$.

This is the pressure acting on the float, and urging the wheel round its axis. The wheel must yield to this motion, if the resistance of the work does not exert a superior pressure on the float in the opposite direction. By yielding, the float withdraws from the impulse, and this is therefore diminished. The wheel accelerates, the resistances increase, and the impulses diminish till they become an exact balance for the resistances. The motion now remains uniform, and the momentum of impulse is equal to that of resistance. The performance of the mill therefore is determined by this; and, whatever be the construction of the mill, its performance is best when the momentum of impulse is greatest. This is had by multiplying the pressure on the float by its velocity. Therefore the momentum will be expressed by

\[F \times \frac{(v-u)^2}{v^2} \times u.\]

But since F and $u^2$ are constant quantities, the momentum will be proportional to $u \times (v-u)^2$. Let x represent the relative velocity. Then $v-x$ will be $= u$, and the momentum will be proportional to $(v-x) \times x^2$, and will be a maximum when $(v-x) \times x^2$ is a maximum, or when $vx^2 - x^3$ is a maximum. This will be discovered by making its fluxion $= 0$; that is,

\[2exdx - 3x^2dx = 0\]

and \[2ex - 3x^2 = 0,\]

or \[2e - 3x = 0\]

and \[2e = 3x,\] and $x = \frac{2e}{3}$; and therefore $v-x$, or $u = \frac{1}{3}v$. That is, the velocity of the float must be one third of the velocity of the stream. It only remains to say what is the absolute pressure on the float thus circumstanced. Let the velocity v be supposed to arise from the pressure of a head of water h. The common theory teaches that the impulse on a given surface S at rest is equal to the weight of a column $hS$; put this in place of F, and $\frac{2}{3}v^2$ in place of $(v-u)^2$ and $\frac{1}{3}v$ for $u$. This gives us $S \times \frac{2}{3}v^2$ for the momentum. Now the power expended is $S \times v$, or the column $S \times h$ moving with the velocity v. Therefore the greatest performance of an undershot wheel is equivalent to raising $\frac{2}{3}$ths of the water that drives it to the same height.

But this is too small an estimation; for the pressure exerted on a plane surface, situated as the float of a mill-wheel, is considerably greater than the weight of the column $S \times h$. This is nearly the pressure on a surface wholly immersed in the fluid. But when a small vein strikes a larger plane, so as to be deflected on all sides in a thin sheet, the impulse is almost double of this. This is in some measure the case in a mill-wheel. When the stream strikes it, it is heaped up along its face, and falls back again, and during this motion it is acting with a hydrostatic pressure on it. When the wheel dips into an open river, this accumulation is less remarkable, because much escapes laterally; but in a mill-course it may be considerable.

We have considered only the action on one float, but several generally act at once. The impulse on most of them must be oblique, and is therefore less than when the same stream impinges perpendicularly; and this diminution of impulse is, by the common theory, in the proportion of the sine of the obliquity. For this reason it is maintained, that the impulse of the whole stream on the lowest float-board, which is perpendicular to the stream, is equal to the sum of the impulses made on all the floats which then dip into the water; or that the impulse on any oblique float is precisely equal to the impulse which that part of the stream would have made on the lowest float-board had it not been interrupted. Therefore it has been recommended to make such a number of float-boards, that when one of them is at the bottom of the wheel, and perpendicular to the stream, the next in succession should be just entering into the water. But since the impulse on a float by no means annihilates all the motion of the water, and it bends round it and hits the one behind with its remaining force, there must be some advantage gained by employing a greater number of floats than this rule will permit. This is abundantly confirmed by the experiments of Smeaton and Bossut. The latter formed three or four suppositions of the number of floats, and calculated the impulse on each, according to the observations made in a course of experiments by the Academy of Sciences, and inserted by us in the article Resistance of Fluids; and when he summed them up, and compared the results with his experiments, he found the agreement very satisfactory. He deduces a general rule, that if the velocity of the wheel is one third of that of the stream, and if seventy-two degrees of the circumference are immersed in the stream, the wheel should have thirty-six floats. Each will dip one fifth of the radius. The velocity being still supposed the same, there should be more or fewer floats according as the arch is less or greater than seventy-two degrees.

Such is the theory, and such are the circumstances which it leaves undetermined. The accumulation of the water on a float-board, and the force with which it may still strike another, are too intricate to be assigned with any tolerable precision: for such reasons we must acknowledge that the theory of undershot wheels is still very imperfect, and that recourse must be had to experience for their improvement. We therefore strongly recommend the perusal of Mr Smeaton's experiments on undershot wheels, contained in the same dissertation with those which we have quoted on overshot wheels. We have only to observe, that to an ordinary reader the experiments will appear too much in favour of undershot wheels. His aim is partly to establish a theory, which will state the relation between their performance and the velocity of the stream, and partly to state the relation between the power expended and the work done. The velocity in his experiments is always considerably below that which a body would acquire by falling from the surface of the head of water; or it is the velocity acquired by a shorter fall. Therefore if we estimate the power expended by the quantity of water multiplied by this diminished fall, we shall make it too small, and the difference in some cases is very great; yet, even with these concessions, it appears that the utmost performance of an undershot wheel does not surpass the raising one third of the expended water to the place from which it came. It is therefore far inferior to an overshot wheel expending the same power; and M. Belidor has led engineers into very mistaken maxims of construction, by saying that overshot wheels should be given up, even in the case of great falls, and that we should always bring on the water from a sluice in the very bottom of the dam, and bring it to the wheel with as great a velocity as possible. Mr Smeaton also says, that the maximum takes place when the velocity of the wheel is two fifths of that of the stream, instead of two sixths according to the theory; and this agrees with the experiments of Bossut. But he measured the velocity by means of the quantity of water which ran past. This must give a velocity somewhat too small, as will appear by attending to Du Buat's observations on the superficial, the mean, and the bottom velocities.

The rest of his observations are most judicious, and well adapted to the instruction of practitioners. We have only to add to them the observations of De Parcieux and Bossut, who have evinced, by very good experiments, that there is a very sensible advantage gained by inclining the float-boards to the radius of the wheel about twenty degrees, so that the lowest float-board shall not be perpendicular, but have its point turned up the stream about twenty degrees. This inclination causes the water to heap up along the float-board, and act by its weight. The floats should therefore be made much broader than the vein of water interrupted by them is deep.

Some engineers, observing the great superiority of overshot wheels above undershot wheels driven by the same expense of power, have proposed to bring the water home to the bottom of the wheel on an even bottom, and to make the float-board no deeper than the aperture of the sluice, which would permit the water to run out. The wheel is to be fitted with a close sole and sides, exactly fitted to the end of this trough, so that if the wheel is at rest, the water may be dammed up by the sole and float-board. It will therefore press forward the float-board with the whole force of the head of water. But this cannot answer; for if we suppose no float-boards, the water will flow out at the bottom, propelled in the manner those persons suppose; and it will be supplied from behind, the water coming slowly from all parts of the trough to the hole below the wheel. But now add the floats, and suppose the wheel in motion with the velocity that is expected. The other floats must drag into motion all the water which lies between them, giving to the greatest part of it a motion vastly greater than it would have taken in consequence of the pressure of the water behind it; and the water out of the reach of the floats will remain still, which it would not have done independently of the float-boards above it, because it would have contributed to the expense of the hole. The motion therefore which the wheel will acquire by this construction must be so different from what is expected, that we can hardly say what it will be.

We are therefore persuaded that the best way of delivering the water on an undershot wheel in a close mill-course is, to let it slide down a very smooth channel, without touching the wheel till near the bottom, where the wheel should be exactly fitted to the course; or to make the floats exceedingly broader than the depth of the vein of water which glides down the course, and allow it to be partly intercepted by the first floats, and heap up along them, acting by its weight after its impulse has been expended. If the bottom of the course be an arch of a circle described with a radius much greater than that of the wheel, the water which slides down will be thus gradually intercepted by the floats.

Attempts have been made to construct water-wheels which receive the impulse obliquely, like the sails of a common wind-mill. This would, in many situations, be a very great acquisition. A very slow but deep river could in this manner be made to drive our mills; and although much power is lost by the obliquity of the impulse, the remainder may be very great. It is to be regretted that these attempts have not been more zealously prosecuted; for we have no doubt of their success in a very serviceable degree. Engineers have been deterred, because when such wheels are plunged in an open stream, their lateral motion is too much impeded by the motion of the stream. We have however seen one which was very powerful. It was a long cylindrical frame, having a plate standing out from it about a foot broad, and surrounding it with a very oblique spiral like a cork-screw. This was plunged about one fourth of its diameter (which was about twelve feet), having its axis in the direction of the stream. By the work which it was performing, it seemed more powerful than a common wheel which occupied the same breadth of the river. Its length was not less than twenty feet: it might have been twice as much, which would have doubled its power, without occupying more of the water-way. Perhaps such a spiral, continued to the very axis, and moving in a hollow canal wholly filled by the stream, might be a very advantageous way of employing a deep and slow stream.

But mills with oblique floats are most useful for employing small streams, which can be delivered from a spout with a great velocity. Bossut has considered these with due attention, and ascertained the best modes of construction. There are two which have nearly equal performances.

1. The vanes being placed like those of a wind-mill, round the rim of a horizontal or vertical wheel, and being made much broader than the vein of water which is to strike them, let the spout be so directed that the vein may strike them perpendicularly. By this measure it will be spread about on the vane in a thin sheet, and exert a pressure nearly equal to twice the weight of a column whose base is the orifice of the spout, and whose height is the fall producing the velocity. Mills of this kind are much in use in the south of Europe. The wheel is horizontal, and the vertical axis carries the millstone; so that the mill is of the utmost simplicity: and