and the resistance of the air; also what is the real velocity of the water at the instant that it strikes the wheel, and the real quantity of water expended in a given time.

"From the velocity of the water, at the instant that it strikes the wheel being given, the height of head productive of such velocity can be deduced, from acknowledged and experimented principles of hydrostatics; so that by multiplying the quantity or weight of water really expended in a given time by the height of a head so obtained, which must be considered as the height from which that weight of water had descended in that given time, we shall have a product equal to the original power of the water, and clear of all uncertainty that would arise from the friction of the water, in passing small apertures, and from all doubts, arising from the different measures of spouting waters, assigned by different authors. On the other hand, the sum of the weights raised by the action of this water, and of the weight required to overcome the friction and resistance of the machine, multiplied by the height to which the weight can be raised in the time given, the product will be equal to the effect of that power; and the proportion of the two products will be the proportion of the power to the effect: so that by loading the wheel with different weights successively, we shall be able to determine at what particular load and velocity of the wheel the effect is a maximum."

The nearer the effect obtained approaches to the power expended in obtaining it, the better and more perfect is the machine. Let the student bear in mind that the effect can never be greater than the cause. This is too often forgotten, and much money has been spent and time wasted in contriving machines to perform impossibilities.

The principles of mechanics are now so generally well understood that perhaps no person into whose hands this book may fall, will ever dream of inventing a machine which shall give a result equal to the motive power and overcome its own friction; yet the time has not long gone by when ingenious persons, reasoning on false premises, vainly flattered themselves that they might accomplish such things and devise some machine which should continue in perpetual motion without a maintaining power.

Mr. Smeaton goes on to show how the velocity of the water striking the wheel may be practically ascertained: first, by running the wheel unloaded in the water, and

then by assisting it by means of counter-weights or cord wound round the axle, until the velocity of the wheel is identical with that of the water and the counterweight, equal to friction and resistance of the air; for, if it were too little, the water would accelerate the wheel beyond the weight; and if too great, retard it; so that the water becomes a regulator of the wheel's motion, and the velocity of its circumference becomes a measure of the velocity of the water. The velocity thus determined, the virtual or effective head may be determined by the law of gravitation; and although, as Mr. Smeaton observed, the virtual head bears no certain or definite proportion to the actual head-as indeed has been shown in a foregoing part of this book-yet, when the aperture is greater, or the velocity of the water issuing therefrom is less, they approach nearer to a coincidence; and consequently, in large openings of mills and sluices, where great quantities of water are discharged from moderate heads, the actual head of water and the vertical head, determined from the velocity, will the more nearly agree, as experience confirms.

From the numerous experiments he had made on the undershot-wheel, Mr. Smeaton deduced the following rules, or, as he calls them, maxims :—

"1. That the virtual, or effective head, being the same, the effect will be nearly as the quantity of water expended. "2. That the expense of water being the same, the effect will be nearly as the height of the virtual or effective head.

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

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

Upon comparing the several proportions between power and effect, remarked during the course of his experiments, Mr. Smeaton observes, the most general is that of 10 to 3, the extremes 10 to 3.2 and 10 to 2.8; but as it appears, that where the quantity of water, or the velocity thereof, that is, where the power is greatest, the second term of the ratio is greatest also, we may therefore well allow the proportion subsisting in large works as 3 to 1.

He also observes, that the proportions of velocity between the water and the wheel are contained in the limits of 3 to 1 and 2 to 1; but as the greater velocities approach the limit 3 to 1, and the greater quantities of water to that of

2 to 1, the best general proportion will be that of 5 to 2. He endeavoured to ascertain what is the ratio between the load such a wheel would carry at the maximum of effect, and what will totally stop it, and found that it would work steadily until that proportion was as 4 to 3; but when this limit was exceeded, the whole worked irregularly, and was liable to be stopped.

The principal aim, however, of a good millwright, is to make the wheel work to the greatest advantage; and the last-mentioned experiments were therefore more curious than useful; yet they are highly interesting, as they make the investigation complete, and anticipate a question which might very naturally be asked.

Mr. Smeaton mentions, that in his working model of an undershot water-wheel, the maximum load was equal to 91b. 6oz., and that the wheel ceased moving with 12lb. in the scale; to which, if the weight of the scale is added, nearly 10oz., the proportion will be nearly as 3 to 4, between the load at the maximum and that by which the wheel is stopped: and he says,

"It is somewhat remarkable, that though the velocity of the wheel, in relation to the water, turns out greater than one-third of the velocity of the water, yet the impulse of the water, in the case of a maximum, is more than double of what is assigned by theory; that is, instead of four-ninths of the column, it is nearly equal to the whole column."

It must be remembered, that in the present case the wheel is not placed in an open river, where the natural current, after it has communicated its impulse to the float, has room on all sides to escape; but in a conduit or race, to which the float-board being adapted, the water cannot otherwise escape than by moving along with the wheel; and when a wheel works in this manner, as soon as the water strikes the float it receives a sudden check, and rises up against the float, like a wave against a fixed object; so that in the working model, when the sheet of water was not a quarter of an inch thick before it met the float, yet this sheet acted against the whole surface of a float three inches high; and, consequently, if the float were no higher than the thickness of the sheet of water, a great part of the force would be lost by the water dashing over the float.

Although in this country the value of water power, and the necessity to make the most of it, has gradually caused the undershot wheel to be abandoned, and the breast wheel

to take its place, whereon the gravity or weight o water acts instead of its impulse; yet there are purposes to which the undershot wheel may be applied

advantage, and to none more than the sawing of timbe especially in new settlements, and in those localities wher water power is abundant and mechanical skill is scarce where labour is expensive and timber costs nothing. I

such circumstances, a simple and efficient saw-mill is of great advantage, and it may be worked at once from the axle of the undershot water-wheel, working two saw frames, by means of cranked arms upon the ends of it; or, if the axle be made of iron, a crank may be formed in it to work a single saw frame, as shown in the annexed woodcut, reduced from an engraving in the Professional Papers of the Royal Engineers, vol. vi., in which will also be found a minute description, elaborately illustrated, of the saw mills and machinery for raising timber in Chatham dockyard, erected by the late Sir Mark Isambard Brunel. (See fig. 26.)

In this engraving A is the dam, frequently formed of square logs, resting against a standard secured by struts, provision being made to carry off the surplus water. the sluice, which, being raised to work the wheel, admits the water into the trough c; here it strikes the float-boards of the wheel D, which is generally made of small diameter, so that the velocity of the water may cause it to make as many revolutions as possible, consistent with the requisite power; the saws making as many strokes as the wheel makes revolutions. E, the crank on the wheel shaft, to which is adapted the connecting rod F, which is attached to the bottom of the saw-frame G. This slides up and down between the standards with an alternating motion, the strokes being double the length of the crank arm. K is the log to be cut; it is mounted on the frame L, which has a rack fixed in its under surface, and is supported by the rollers a a.

The pinion b on the axis of the wheel м works in the rack, and according as the wheel moves forward or backward, it works the frame, moving it towards or away from the saws. Motion is given to this wheel by the pall c, the other end of which is jointed to one of the sides of the arm of the bent lever d. This lever is moved backward and forward by the rod e which is jointed to the bent rod f, and this rod, or rather lever, is fitted upon an axle attached at one end to the frame of the wooden building and at the other to the frame of the saw-mill. When it is requisite to reverse the motion, after the log is cut, the pall c is lifted clear of teeth of the ratchet-wheel M; and this wheel is turned in an opposite direction by hand.

A saw-mill on this principle was made by the late Mr. Rennie, of which he has given the following brief