WATER-WORKS.

tity ZBD; 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 concur 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 40 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 1/3th more than the depth GH of the shrouding. The start AB is 2/3 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 2/3ths of GH or AI. Then CD is so placed that HD is about 1/3th 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 1/3th of the diameter from the bottom, and half of the water will be discharged from the lowest bucket, about 1/2th 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 1/3th (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 1/3th of the radius (being 0.18285), or 1/3th of the diameter. It is evident, that if only 1/3 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 admission 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, 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.

A bucket wheel was some time ago executed by Mr Robert Burns, at the cotton mills of Houston, Burns, and Co. at Cartside in Renfrewshire, 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 farther 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 15° from the bottom; and they retain one 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 spout (also the invention of Mr Burns, and furnished with a rack-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 carry the performance as far as it can go. Mr Burns made the first trial on a wheel of 24 feet diameter; and its performance is manifestly superior to that of the wheel which it replaced, and which was a very good one. It has also another valuable property: When the supply of water is very scanty, a proper adjustment of the apparatus in the spout will direct almost the whole of the water into the outer buckets; which, by placing it at a greater distance from the axis, makes a very sensible addition to its mechanical energy.

We 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 this series of partitions must greatly contribute to the stillness 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. This 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 ratchets, 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 300 times rarer than water, will escape through a hole almost 30 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 14 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, which rendered the motion very hobbling. When three holes of an inch diameter were made in each bucket (12 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... 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 our 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 our fall is exceedingly great, a wheel of an equal diameter becomes enormously big and expensive, and is of itself an unmanageable load. We have seen wheels of 58 feet diameter, however, 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 that any fall of water whatever can be applied, and 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 big enough and light enough, they would buoy and even draw up the plugs 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. Mr 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 wa-

ter 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, which 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 equilibrio 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 reaction; and in this balance gravity and the reacting 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 peritrocheo), 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 spout 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 A f H (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 AP. 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, sum, and uniformly distributed in the space BER \( \frac{1}{3} \). This constitutes a ring of uniform thickness. Let the area of its cross section \( \beta B \) and \( F f \) 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 \( b r r b \) pressing on \( F f \), or acting by the lever \( C F \). The weight of this pillar may be expressed by \( a \times b r \), 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 \( R : r = v : u \). 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 best 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. If the wheel moves fast, little water lies in each bucket, and \( a \) 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 up as possible. But this diminishes \( AP \), and this diminishes the velocity of the wheel: and as this has no limit, the proposition is demonstrated; and an overshot wheel does the more work as it moves slowest.

Convincing as this discussion must be to any mechanician, 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 adding a popular view of the question, which requires no such investigation.

We may reason in this way: Suppose a wheel having 30 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 30 buckets, to the same height, and with the same velocity. Suppose, farther, that when the load on the rising side of the machine is one half of that on the wheel, the wheel makes four turns in a minute, or one turn in 15 seconds. During this time 92 cubic feet of water have flowed into the 30 buckets, and each has received three cubic feet. Then each of the rising buckets contains \( \frac{1}{3} \) feet; and 45 cubic feet are delivered into the upper cistern during one turn of the wheel, and 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 30 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 90 cubic feet during each turn, and 360 in a minute. But, by supposition, it is making but two turns in a minute; this must proceed from a greater load than three cubic feet of water in each rising bucket. The machine must therefore be raising more than 90 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 slow as before, there is more than twice the former quantity in the 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 farther, and assert, that the more we retard the machine, by loading it with more work of a similar kind, the greater will be its performance. This does not immediately appear from the present discussion: But let us call the first quantity of water in the rising bucket \( A \); the water raised by four turns in a minute will be \( 4 \times 30 \times A = 120 A \). 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 \( 2 \times 30 \times 2A + x = 120 A + 60x \). 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 \( 32 + 4A + 2x + y = 120 A + 60x + 30y \).

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. water in the buckets. Our mill-wrights 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 \( FC : AC = \text{gravity} : \text{centrifugal force} \).

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 \( \alpha \) equal to AL, and the arch \( \beta \) equal to AM: Then C and C will be the positions of the bucket on the revolving wheel, corresponding to CDO and CEO on the wheel at rest. Water will begin to run out at \( \alpha \), and it will be all gone at \( \beta \).—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 confidential conclusions from any experiments that can be made with them; and it is therefore no matter of wonder that the opinions of mill-wrights 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 mill-wrights 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. Mr Euler says, that the performance of the best mill cannot exceed that of the worst above \( \frac{1}{4} \)th: 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 carpenters 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 not 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 Karstner of Gottingen; 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 valuable paper on this subject by Mr Lambert, in the Memoirs of the Academy of Berlin for the year 1775. From these dissertations, and from the Hydrodynamique of the abbe Bossut, the reader will get 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 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 in- crease by the changes of form occasioned by the wearing of the machine—much greater irregularities arise from the subsultory motions of cranks, stamper, and other parts which move unequally or reciprocally. A machine may be so loaded as just to be in equilibrium 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 get the best information of their resistances. The comparison with mills worked by Watt and Boulton's steam-engines is perhaps a better measure of the resistances opposed by different kinds of work, because their power is very distinctly known. We have been informed by one of the most eminent engineers, that a ton and a half of water per minute falling one foot will grind and dress one bushel of wheat per hour. This is equivalent to 9 tons falling 10 feet.

If an overshot-wheel opposed no resistance, and only one bucket were filled, the wheel would acquire the velocity due to a fall through the whole height. But when it is in this state of accelerated motion, if another bucket of water is delivered into it, its motion must be checked at the first, by the necessity of dragging forward this water. If the 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, the 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 intended 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 this impulse; because the water falls among other water, or it strikes the boards of the wheel with such obliquity that it cannot produce any such effect.

It is a much debated question among millwrights, whether the diameter of the wheel should be such as that the water will be delivered at the top of the wheel? or larger, so 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 loss of fall 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 insert the determination that we have given, of the loss of power produced by the centrifugal force. But even with this in 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 bit of a stick or stone falling in about the wheel tears off part of the shrouding or bucket, and frosty weather frequently binds all fast. It therefore seldom answers expectations. We have nothing to add on this case. to what we have already extracted from Mr Smeaton's Dissertation on the Subject of Breast or half Overshot Wheels.

There is another form of wheel by which water is made to act on a machine by its weight, which merits consideration. This 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, Bc, which have a hole C c 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 c 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 utmost 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 this 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 hydraulicist), 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 2/3ths 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 AH h a 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 AC c a and the inner conoid HG g h 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 CG, c g. On the outer circumference of this bottom are 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 reaction of fluids, this should be a very powerful machine, and should raise 2/3ths of the water expended. But if we admit the re-action action 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 any thing of this kind in his *Exercitationes Hydraulicae*, where he particularly describes the machine. This remissness probably has proceeded from fixing the attention on Euler's improved construction. It is plain that this must be a most burdensome 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 an additional argument taken from an addition made to it by Mr 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 1½; 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. This gives a velocity of 46 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 37½. 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 17 times the weight of the water in the orifice.

The empty machine weighed 80 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 CE c c 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 C c 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 LKM m k l. This is kept in its place by means of a metal or wooden ring N n, thin at the upper edge, and taper-shaped. This is drawn in above the leather, and stretches it, and 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 at the small crevice, between the two metal shoulders. These shoulders need not touch, so that the friction must be insensible. We imagine that this method of tightening a turning joint may be used with great advantage in many cases.

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

2. Of Undershot Wheels.

All wheels go by this name where the motion of the water is quicker than that of the partitions or boards of the wheel, and it therefore impels them. These are called the float boards, 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 dynamical and mathematical discussion, have hardly done anything more than copied the earliest deductions from the simple theory of the resistance of fluids. The consequence has been, that our practical knowledge is very imperfect; and it is still chiefly from experience that we must learn the performance of undershot wheels. Unfortunately this stops their 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 ascribe 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 repeat here the defects of the common explanations of the motions of undershot wheels. The part of this 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 held fast to its impulse \( f \) on the float moving with the velocity \( u \), as \( v^2 \) to \( v-u^2 \), and \( f = F \times \frac{v-u^2}{v} \).

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

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

But since \( F \) and \( u^2 \) are constant quantities, the momentum will be proportional to \( u \times \frac{v-u^2}{v} \). 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 \( v-x^2 - x^3 \) is a maximum. This will be discovered by making its fluxion \( = 0 \). That is,

\[ 2v \times x - 3x^2 = 0, \] and \[ 2v \times x - 3x^2 = 0. \]

or \[ 2v = 3x, \text{ and } x = \frac{v}{3}. \]

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 \( S \times h \); put this in place of \( F \), and \( \frac{1}{2} v^2 \) in place of \( v-u^2 \) for \( u \). This gives us \( S \times \frac{1}{2} v \) 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 \( \frac{1}{2} \) 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 floatboard, 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 floatboard had it not been interrupted. Therefore it has been recommended to make such a number of floatboards, 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 Bosset. Mr Bosset 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 made 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 72 degrees of the circumference are immersed in the stream, the wheel should have 36 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 72 degrees.

Such is the theory, and such are the circumstances which it leaves undetermined. The accumulation of the water on a floatboard, 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 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 ex- pended by the quantity of water multiplied by this di- minished fall, we shall make it too small; and the dif- ference 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 expend- ing the same power; and Mr 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 Bosset. 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 attend- ing to 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 Des Par- cieux and Bossut, who have evinced, by very good ex- periments, that there is a very sensible advantage gained by inclining the floatboards to the radius of the wheel about 20 degrees, so that the lowest floatboard shall not be perpendicular, but have its point turned up the stream about 20 degrees. This inclination causes the water to heap up along the floatboard, 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 wa- ter home to the bottom of the wheel on an even bot- tom, and to make the floatboard no deeper than the ap- erture 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 floatboard. It will therefore press forward the floatboard with the whole force of the head of water. But this cannot answer; for if we suppose no floatboards, 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 ve- locity that is expected. The other floats most drag in- to motion all the water which lies between them, giv- ing 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 independent of the floatboards 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 de- livering 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 im- pulse 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 ri- ver 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 zeal- ously prosecuted; for we have no doubt of their suc- cess 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 seen one, how- ever, which was very powerful: It was a long cylin- drical 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 12 feet), hav- ing 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 20 feet: it might have been twice as much, which would have doubled its power, without occupying more of the wa- ter-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 employ- ing a deep and slow stream.

But mills with oblique floats are most useful for em- ploying small streams, which can be delivered from a spout with a great velocity. Mr Bossut has considered these with due attention, and ascertained the best modes of construction. There are two which have nearly e- qual 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 di- rected 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 axle carries the millstone; so that the mill is of the utmost simplicity: and this is its chief recommendation; for its power is greatly inferior to that of a wheel construc- ted in the usual manner.

2. The vanes may be arranged round the rim of the wheel... wheel, not like the sails of a wind-mill, but in planes inclined to the raili, but parallel to the axis, or to the planes passing through the axis. They may either stand on a sole, like the oblique floats recommended by De Pareicoux, as above mentioned; or they may stand on the side of the rim, not pointing to the axis, but aside from it.

This disposition will admit the spout to be more conveniently disposed either for a horizontal or a vertical wheel.

We shall conclude this article by describing a contrivance of Mr Burns, the inventor of the double bucketed wheel, for fixing the arms of a water-wheel. It is well known to millwrights that the method of fixing them by making them to pass through the axle, weakens it exceedingly, and by lodging water in the joint, soon causes it to rot and fail. They have, therefore, of late years put cast-iron flanges on the axis, to which each arm is bolted; or the flanges are so fashioned as to form boxes, serving as mortises to receive the ends of the arms. These answer the purpose completely, but are very expensive; and it is found that arms of fir bolted into flanges of iron, are apt to work loose. Mr Burns has made wooden flanges of a very curious construction, which are equally firm, and cost much less than the iron-ones.

This flanch consists of eight pieces, four of which compose the ring represented in fig. 15, meeting in the joints \(a_b, a_b, a_b, a_b\), directed to the centre O. The other four are covered by these, and their joints are represented by the dotted lines \(a_b, a_b, a_b, a_b\). These two rings break joint in such a manner that an arm MN is contained between the two nearest joints \(a_b'\) of the one, and \(a_b'\) of the other. The tenon formed on the one end of the arm A, &c., is of a particular shape: one side, GF, is directed to the centre O; the other side, BCDE, has a small shoulder BC; then a long side CD directed to the centre O; and then a third part DE parallel to GF, or rather diverging a little from it, so as to make up at E the thickness of the shoulder BC; that is, a line from B to E would be parallel to CD. This side of the tenon fits exactly to the corresponding side of the mortise; but the mortise is wider on the other side, leaving a space GFK a little narrower at FK than at GH. These tenons and mortises are made extremely true to the square; the pieces are put round the axle, with a few blocks or wedges of soft wood put between them and the axle, leaving the space empty opposite to the place of each arm, and firmly bolted together by bolts between the arm-mortises. The arms are then put in, and each is pressed home to the side CDE, and a wedge HF of hard wood is then put into the empty part of the mortise and driven home. When it comes through the flanch and touches the axle, the part which has come through is cut off with a thin chisel, and the wedge is driven better home. The spaces under the ends of the arms are now filled with wedges, which are driven home from opposite sides, till the circle of the arms stands quite perpendicular on the axle, and all is fast. It needs no hoops to keep it together, for the wedging it up round the axle makes the two half rings draw close on the arms, and it cannot start at its own joints till it crushes the arms. Hoops, however, can do no harm, when all is once wedged up, but it would be improper to put them on before this be done.

A very curious hydraulic machine was erected at Zurich by H. Andreas Wirtz, a tinplate worker of that place. The invention shows him to be a person of very uncommon mechanical knowledge and sagacity. As it is a machine which operates on a principle widely different from all other hydraulic machines, and is really excellent in its kind, we presume that our readers will not be displeased with some account of it.

Fig. 16. is a sketch of the section of the machine, as Fig. 16. it was first erected by Wirtz at a dyehouse in Limmat, in the suburbs or vicinity of Zurich. It consists of a hollow cylinder, like a very large grindstone, turning on a horizontal axis, and partly plunged in a cistern of water. The axis is hollow at one end, and communicates with a perpendicular pipe CBZ, part of which is hid by the cylinder. This cylinder or drum is formed into a spiral canal by a plate coiled up within it like the main-spring of a watch in its box; only the spires are at a distance from each other, so as to form a conduit for the water of uniform width. This spiral partition is well joined to the two ends of a cylinder, and no water escapes between them. The outermost turn of the spiral begins to widen about three-fourths of a circumference from the end, and this gradual enlargement continues from Q to S nearly a semicircle; this part may be called the Horn. It then widens suddenly, forming a Scoop or shovel SS'. The cylinder is supported so as to dip several inches into the water, whose surface is represented by VV'.

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

After the scoop has emerged, the water passes along the spiral by the motion of it round the axis, and drives the air before it into the rising-pipe, where it escapes.—In the mean time, air comes in at the mouth of the scoop; and when the scoop again dips into the water, it again takes in some. Thus there is now a part filled with water and a part filled with air. Continuing this motion, we shall receive a second round of water and another of air. The water in any turn of the spiral will have its two ends on a level; and the air between the successive columns of water will be in its natural state; for since the passage into the rising-pipe or MAIN is open, there is nothing to force the water and air into any other position. But since the spires gradually diminish in their length, it is plain that the column of water will gradually occupy more and more of the circumference of each. At last it will occupy a complete turn of some spiral that is near the centre; and when sent farther in, by the continuance of the motion, some of it will run back over the top of the succeeding spiral. Thus it will run over at K 4 into the right-hand side of the third spiral. Therefore it will push the water of this spire backwards, and raise its other end, so that it also will run over backwards before the next turn be completed. And this change of disposition will at last reach the first or outermost spiral, and some water will run over into the horn and scoop, and finally into the cistern.

But as soon as water gets into the rising-pipe, and rises rises a little in it; it stops the escape of the air when the next scoop of water is taken in. Here are now two columns of water acting against each other by hydrostatic pressure and the intervening column of air. They must compress the air between them, and the water and air columns will now be unequal. This will have a general tendency to keep the whole water back, and cause it to be higher on the left or rising side of each spire than on the right descending side. The excess of height will be just such as produces the compression of the air between that and the preceding column of water. This will go on increasing as the water mounts in the rising-pipe; for the air next to the rising-pipe is compressed at its inner end with the weight of the whole column in the main. It must be as much compressed at its outer end. This must be done by the water column without it; and this column exerts this pressure partly by reason that its outer end is higher than its inner end, and partly by the transmission of the pressure on its outer end by air, which is similarly compressed from without. And thus it will happen that each column of water, being higher at its outer than at its inner end, compresses the air on the water column beyond or within it, which transmits this pressure to the air beyond it, adding to it the pressure arising from its own want of level at the ends. Therefore the greatest compression, viz. that of the air next the main, is produced by the sum of all the transmitted pressures; and these are the sum of all the differences between the elevation of the inner ends of the water columns above their outer ends: and the height to which the water will rise in the main will be just equal to this sum.

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

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

But when the main is very high, this construction will require a very great diameter of the drum, or many turns of a very narrow pipe. In such cases it will be much better to make the spiral in the form of a corkscrew, as in fig. 17, instead of this flat form like a watch-spring. The pipe which forms the spiral may be lapped round the frustum of a cone, whose greatest diameter is to the least (which is next to the rising-pipe) in the same proportion that we assigned to CB and CL. By this construction the water will stand in every round so as to have its upper and lower surfaces tangents to the top and bottom of the spiral, and the water columns will occupy the whole ascending side of the machine, while the air occupies the descending side.

This form is vastly preferable to the flat: it will allow us to employ many turns of a large pipe, and therefore produce a great elevation of a large quantity of water. The same thing will be still better done by lapping the pipe on a cylinder, and making it taper to the end, in such a proportion that the contents of each round may be the same as when it is lapped round the cone. It will raise the water to a greater height (but with an increase of the impelling power) by the same number of turns, because the vertical or pressing height of each column is greater.

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

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

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

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

The circumference of the first spire is \( \frac{1}{m} + 1 \) or 2. Let D and d be the diameters of the first and last spires; we have \( \frac{2}{m} : \frac{1}{m} = D : d \), or \( \frac{2m}{m} : \frac{1}{m} = D : d \).

Therefore if a pipe of uniform bore be lapped round a cone, of which D and d are the end diameters, the spirals will be very nearly such as will answer the purpose. It will not be quite exact, for the intermediate spirals will be somewhat too large. The conoidal frustum should be formed by the revolution of a curve of the logarithmic kind. But the error is very trifling.

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

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

Let ABEO (fig. 18.) represent the first or outermost round of the spiral, of which the axis is C. Suppose it immersed up to the axis in the water VV', we have seen that the machine is most effective when the surfaces KB and ON of the water columns are distant the whole diameter BO of the spiral. Therefore let the pipe be first supposed of equal caliber to the very mouth Ec, which we suppose to be just about to dip into the water. The surface ON is kept there, in opposition to the pressure of the water column BAO, by the compressed air contained in the quadrant OE, and in the quadrant which lies behind EB. And this compression is supported by the columns behind, between this spire and the rising pipe. But the air in the outermost quadrant EB is in its natural state, communicating as yet with the external air. When, however, the mouth Ec has come round to A, it will not have the water standing in it in the same manner, leaving the half space BEO filled with compressed air; for it took in and confined only what filled the quadrant BE. It is plain, therefore, that the quadrant BE must be so shaped as to take in and confine a much greater quantity of air; so that when it has come to A, the space BEO may contain air sufficiently dense to support the column AO. But this is not enough: For when the wide mouth, now at A, rises up to the top, the surface of the water in it rises also, because the part AO OA is more capacious than the cylindric part OE EO which succeeds it, and which cannot contain all the water that it does. Since, then, the water in the spire rises above A, it will press the water back from ON to some other position m' n', and the pressing height of the water column will be diminished by this rising on the other side of O. In short, the horn must begin to widen, not from B, but from A, and must occupy the whole semicircle ABE; and its capacity must be to the capacity of the opposite cylindrical side as the sum of BO, and the height of a column of water which balances the atmosphere to the height of that column. For then the air which filled it, when of the common density, will fill the uniform side BEO, when compressed so as to balance the vertical column BO. But even this is not enough; for it has not taken in enough of water. When it dipped into the cistern at E, it carried air down with it, and the pressure of the water in the cistern caused the water to rise into it a little way; and some water must have come over at B from the other side, which was drawing narrower. Therefore when the horn is in the position EOA, it is not full of water. Therefore when it comes into the situation OAB, it cannot be full nor balance the air on the opposite side. Some will therefore come out at O, and rise up through the water. The horn must therefore, 1st, Extend at least from O to B, or occupy half the circumference; and, 2dly, It must contain at least twice as much water as would fill the side BEO. It will do little harm though it be much larger; because the surplus of air which it takes in at E will be discharged, as the end Ec of the horn rises from O to B, and it will leave the precise quantity that is wanted. The overplus water will be discharged as the horn comes round to dip again into the cistern. It is possible, but requires a discussion too intricate for this place, to make it of such a size and shape, that while the mouth moves from E to B, passing through O and A, the surface of the water in it shall advance from E to ON, and be exactly at O when the beginning or narrow end of the horn arrives there.

We must also secure the proper quantity of water. When the machine is so much immersed as to be up to the axis in water, the capacity which thus secures the proper quantity of air will also take in the proper quantity of water. But it may be erected so as that the spirals shall not even reach the water. In this case it will answer our purpose if we join to the end of the horn a scoop or shovel QRSB (fig. 19.), which is so formed as to take in at least as much water as will fill the horn. This is all that is wanted in the beginning of the motion along the spiral, and more than is necessary when the water has advanced to the succeeding spire; but the overplus is discharged in the way we have mentioned. At the same time, it is needless to load the machine with more water than is necessary, merely to throw it out again. We think that if the horn occupies fully more than one half of the circumference, and contains as much as will fill the whole round, and if the scoop lifts as much as will certainly fill the horn, it will do very well.

N. B. The scoop must be very open on the side next the axis, that it may not confine the air as soon as it enters the water. This would hinder it from receiving water enough. The following dimensions of a machine erected at Florence, and whose performance corresponded extremely well with the theory, may serve as an example.

The spiral is formed on a cylinder of 10 feet diameter, and the diameter of the pipe is 6 inches. The smaller end of the horn is of the same diameter; and it occupies three-fourths of the circumference, and it is 7½ inches wide at the outer end. Here it joins the scoop, which lifts as much water as fills the horn, which contains 4340 Swedish cubic inches, each = 1.577 English. The machine makes six turns in a minute, and raises 1354 pounds of water, or 22 cubic feet, 10 feet high in a minute.

The above account will, we hope, sufficiently explain the manner in which this singular hydraulic machine produces its effect. When every thing is executed by the maxims which we have deduced from its principles, we are confident that its performance will correspond to the theory; and we have the Florentine machine as a proof of this. It raises more than ¾ths of what the theory promises, and it is not perfect. The spiral is of equal caliber, and is formed on a cylinder. The friction is so inconsiderable in this machine, that it need not be mended: but the great excellency is that whatever imperfection there may be in the arrangement of the air and water columns, this only affects the elegance of the execution, causing the water to make a few more turns in the spiral before it can mount to the height required; but wastes no power, because the power employed is always in proportion to the sum of the vertical columns of water in the rising side of the machine; and the height to which the water is raised by it is in the very same proportion. It should be made to move very slow, that the water be not always dragged up by the pipes, which would cause more to run over from each column, and diminish the pressure of the remainder.

If the rising-pipe be made wide, and thus room be made for the air to escape freely up through the water, it will rise to the height assigned; but if it be narrow, so that the air cannot get up, it rises almost as slow as the water, and by this circumstance the water is raised to a much greater height mixed with air, and this with hardly any more power. It is in this way that we can account for the great performance of the Florentine machine, which is almost triple of what a man can do with the finest pump that ever was made: indeed the performance is so great, that one is apt to suspect some inaccuracy in the accounts. The entry into the rising-pipe should be no wider than the last part of the spiral; and it would be advisable to divide it into four channels by a thin partition, and then to make the rising-pipe very wide, and to put into it a number of slender rods, which would divide it into slender channels that would completely entangle the air among the water. This will greatly increase the height of the heterogeneous column. It is surprising that a machine that is so very promising should have attracted so little notice. We do not know of any being erected out of Switzerland except at Florence in 1778. The account of its performance was in consequence of a very public trial in 1779, and honourable declaration of its merit, by Sig. Lorenzo Ginori, who erected another, which fully equalled it. It is shortly mentioned by Professor Sulzer of Berlin, in the Sammlungen Vermischten Schriften for 1754.

A description of it is published by the Philosophical Society of Zurich in 1766, and in the descriptions published by the Society in London for the encouragement of arts in 1766. The celebrated Daniel Bernoulli has published a very accurate theory of it in the Petersburg Commentaries for 1772, and the machines at Florence were erected according to his instructions. Baron Alstromer in Sweden caused a glass model of it to be made, to exhibit the internal motions for the instruction of artists, and also ordered an operative engine to be erected; but we have not seen any account of its performance. It is a very intricate machine in its principles; and an ignorant engineer, nay the most intelligent, may erect one which shall hardly do anything; and yet, by a very trifling change, may become very powerful. We presume that failures of this kind have turned the attention of engineers from it; but we are persuaded that it may be made very effective, and we are certain that it must be very durable. Fig. 10, is a section of the manner in which the author has formed the communication between the spiral and the rising pipe. P is the end of the hollow axis which is united with the solid iron axis. Adjoining to P, on the under side, is the entry from the last turn of the spiral. At Q is the collar which rests on the supports, and turns round in a hole of bell-metal. ff is a broad flanch cast in one piece with the hollow part. Beyond this the pipe is turned somewhat smaller, very round and smooth, so as to fit into the mouth of the rising pipe, like the key of a cock. This mouth has a plate ee attached to it. There is another plate dd, which is broader than ee, and is not fixed to the cylindrical part, but moves easily round it. In this plate are four screws, such as g, g, which go into holes in the plate ff, and thus draw the two plates ff and dd together, with the plate ee between them. Pieces of thin leather are put on each side of ee; and thus all escape of water is effectually prevented, with a very moderate compression and friction.