toothed wheels fixed to the shrouding. As applied to common breastwheels adapted for falls not exceeding 18 or 20 feet, these ventilating buckets effect so great an improvement, that if the wheel is plunged in back-water to a depth of 5 or 6 feet, its uniform speed is not impeded. In these wheels the sole of the buckets is close, and the tail end of them being turned up at a distance of 2 inches from the back of the sole-plate, and running parallel with it, terminate within about 2 inches of the bend of the bucket, immediately above it. The water in entering the bucket drives the air out through the aperture into the space behind, and thence into the bucket above, and so on in succession. The converse occurs when the buckets are emptied, as the air is enabled to enter as fast as the wheel arrives at such a position as to permit the water to escape, (For a more copious description see Water-wheels with ventilating buckets.)

There are many cases in which it is of importance to know the proportion of power necessary to give different degrees of velocity to a mill; but as the construction of mills and the purposes they serve are various, it is perhaps impossible to find any law of universal application. Mr. Banks, in his Treatise on Mills,' has drawn a conclusion which he appears to consider invariable, namely, that "when a wheel acts by gravity, its velocity will be as the cube root of the quantity of water it receives."

6

But supposing a wheel to raise water by means of cranks and pumps on Mr. Banks's principle, Buchanan thought it might easily be demonstrated, that by reducing the velocity of the wheel to a certain degree, the wheel would raise more water than would be necessary to move it at that velocity,-a thing evidently impossible.

In this view it would seem tnat there is no actual case in which Mr. Banks's conclusions will hold true. But, however they may apply to other mills, the experiments of Buchanan seem to prove at least that they do not apply to cotton-mills. On the ground of

some experiments made at different times, and with all the attention possible, did he presume to call in question an authority for which the highest respect is entertained.

In January, 1796, he measured the quantity of water the Rothesay old cotton-mill required; first, when going at its common velocity, and secondly, when going at half that velocity. The result was, that the last required just half the quantity of water which the first did. It is to be observed, that in these experiments the quantities of water were calculated from the heads of water and apertures of the sluices.

From these experiments he inferred, "that the quantity of water necessary to be employed in giving different degrees of velocity to a cotton-mill must be nearly as that velocity."

He was satisfied with this experiment, and the inference drawn from it, till some gentlemen well acquainted with the theory and practice of mechanics expressed their doubts on the subject. He had then recourse to another experiment, which he considered less liable to error than the former.

The water which drives the old cotton-mill falls a little below it into a perpendicular-sided pond, which serves as a dam for a cornmill at some distance below it. To ascertain, therefore, the proportional quantities of water used by the old mill, nothing more was necessary than to measure the time the water took to rise to a certain height in that pond; and accordingly, on the 1st of May, 1798, he

The result of these experiments approaches very nearly to that of 1796. The difference may be accounted for by the small degree of leakage which must have taken place at the sluices on the lower end of the pond; and the time being greater in the third and fourth experiments, the leakage would of course be greater.

Smeaton and others have proved in a very satisfactory manner, that "the mechanical power that must of necessity be employed in giving different degrees of velocity to the same body, must be as the square of that velocity." But it appeared to Buchanan, that the result of the above experiments may be easily reconciled to this proposition, by considering what Smeaton says immediately afterwards :-" If the converse of this proposition did not hold true, viz. that if a body in motion, in being stopped, would not produce a mechanical effect equal or proportional to the square of its velocity, or to the mechanical power employed in producing it, the effect would not correspond with its producing cause." It is to be observed, that Smeaton's experiments were

made on the velocity of heavy bodies, free from friction and other causes of resistance; but in mills there is not only friction, but obstacles, to be removed: and experiments made on friction have proved that the friction of many kinds of bodies increases in direct proportion to their velocity. But the velocity of a cotton-mill at work may be considered as a mechanical effect; and, if so, must correspond with its producing

cause.

The preceding experiments on the Rothesay mill are undoubtedly correct and consistent with the principles of motion and power, and also with the experiments of Smeaton on mills and mechanical power.

The mechanical power is as the quantity of water on the wheel, multiplied into its velocity when the wheel, fall, and other circumstances remain the

same; and since the mechanical effect is measured by the resistance multiplied into the velocity of the working point when the friction is constant, if the quantity of water be diminished by its half, either half the resistance, or half the velocity with which it is overcome, must be taken away, otherwise there will not be an equilibrium between the power and effect. But at the same time it is to be observed, than an increased velocity lessens the friction of the intermediate machinery, and consequently a greater effect would be produced by the greater velocity, as appears to be the case by the experiments. There is not, however, in the detail of these experiments, sufficient data by which it becomes easy to arrive at any useful conclusion.

Roberton, an engineer of some eminence, made observations on these experiments, alleging that the conclusions of Banks give most satisfactory evidence that particular care and judgment are necessary in

making such trials. It appeared to Roberton, that the wrong conclusions which have been drawn by many writers on this subject, have wholly arisen from misapprehending some of Sir Isaac Newton's fundamental principles of mechanics, and from a love of establishing theoretical expressions rather than strict observations of the invariable laws of nature,-expressions such as these, viz. quantity of motion, instantaneous impulse.

Taking a constant portion of time (viz. a second) to be the measure of a body, and an instant to be measure of the effect it produces; or by taking time as the measure of the cause, and space as the measure of the effect,-as to an instantaneous effect, Roberton argues that it is an absurdity in itself, as well as in mechanics,we can form no idea of a body put into motion, without the acting power or body act upon the body put into motion for some time, and also over some space; and to suppose otherwise, leads us entirely out of the sound principles of mechanics.

In mechanics every effect is equal to its producing cause. In the case ofa power acting on a body producing motion, and also this body acting against another power which retards its motion, if the causes of action and resistance are each measured by the time the motions are produced and retarded, the result will be equal.

If they be measured by the space over which they act, the results will be equal; and this is an universal principle, whether applied to accelerating power and motion, as gravity, &c., or to machines which act constantly and uniformly. Yet, in the case of uniform motion, space or time may be used at pleasure; as from the uniformity of space and time they become a common measure.

To illustrate this, suppose the

A

body A acted upon by the power of gravity through the space AB, in a portion of time which we will call one. When it arrives at B, it meets with another medium of resistance, which is ten times greater than the former: the body A will be resisted in proportion to the cause of action and resistance; that is to say, if the time of action were one second, the time of resistance will be one-tenth of a second, and the distance A B will be to the distance BC as ten to one; so that whether space or time be taken as the measure of action, the same must be taken for the measure of the effect, to have the results proportionate and equal. But if the cause be measured by time, and the effect by space, the results will be as the squares of the time, or, which is the same thing, as the squares of the velocity.

B

C

Thus, suppose a body in motion, with a velocity of one, has a power to penetrate into a bank of earth 1 foot: if the same body, with a velocity of two, strike the bank, it will penetrate to the depth of feet; for the velocity is double, and the time of action is double, and therefore the results will be compounded of both, that is, as the square of the velocity.

From the above it may be inferred that if equal bodies be acted upon by unequal powers, the time requisite to produce an equal motion will be reciprocally proportionate to the powers; that is to say, if a power of ten act upon a body for one second of time, and the power of one act upon an equal body for ten seconds, they will produce equal velocities. But the spaces through which the bodies are carried are very unequal, being as

ten to one; and if the square roots of the powers producing the effects be taken, that will give the times they take in carrying the body acted upon through equal spaces.

But it is obvious this doctrine has no more to do with the operation of machines than to supply their first starting from rest to the motion necessary for working. When this is acquired, the power applied and the power of resistance balance each other, and whatever be the motion the machine moves at, the same power will carry it on, (if it be upheld,) provided the machine act in such a manner as not to accumulate resistance by the accumulation of motion, which is the case in forcing fluids through pipes, &c. In cases of this kind, the nature of the machine must be particularly kept in view, and no law whatever adopted to explain the resistance the acting body meets with, but what is simply deduced from the very machine which is under consideration; but, in most cases, any machine may be considered as acting purely on a statical principle. The raising of weights, or overcoming friction, Roberton considers purely as acting on that principle; and when the power of action is equal to the resisting power, the machine is indifferent to motion or rest. If the machine be at rest, the power will not move it, being a balance to the resistance. If the machine be set in motion, the power will keep it in the same motion, (provided the power be upheld,) the same as equal weights hung over a pulley, or in the opposite scales of a beam. If they be at rest, they will remain so; and if they be put in motion, they will endeavour to persevere in

the same.

The above doctrine of a statical principle is proved in the most satisfactory manner by the experiments made at the old mill of Rothesay, the motion of the water

wheel being exactly proportional to the quantity of water expended, and therefore an exact and equal load upon the wheel; that is to say, the buckets were equally full when the mill moved at its ordinary motion, or at half that motion.

The effect, therefore, of letting more water on a wheel is not to lodge a greater quantity in the buckets, but to supply the same quantity when the wheel is in a greater motion.

Banks, however, made his experiments agree with his theory, yet Roberton took no trouble in inquiring into them, alleging it would be to little purpose to have done so.

"Suffice it to say," he adds, "that the very small quantities of water which Banks made use of, and the slowness of the motion of his wheel in his experiments, give no ground for placing the smallest dependence on them; and when compared with the more judicious and accurate experiments of Smeaton, they dwindle into contempt." Roberton further says, that "Smeaton, in running his wheel at nearly 3 feet in the second, brought nearly to a maximum, and lost but about one-fourth or one-fifth of the original effect (alluding to his overshot wheels). Banks, at his highest motion, run his wheel about 1 foot in the second, and reducing it to one-half of that motion, the same quantity of water then expended was capable of performing four times the work; and by deduction from thence, it appears plain that his wheel (from his own theory) would perform about twenty times the quantity of work which Smeaton's could perform with the same quantity of water, and about sixteen times more than nature; so that the observation (alluding to the theory of Banks) is very just in saying that, by reducing the mo

tion of the wheel, it is demonstrable it would raise more water than supply itself." Water-wheels (Overshot). The best water-wheel is that which is calculated to produce the greatest effect when it is supplied by a stream furnishing a given quantity of water with a given fall.

The mechanical effect depends on the proportion of the wheel's diameter to the height of the fall, and on the velocity of the circumference of the wheel. These are

the two principal parts to be considered in the theory of wheels, but there are also some other points which ought to be attended to, because the effect is much decreased when they are neglected.

Of the proportion of the radius of the water-wheel to the height of the fall. Let A B C D be the wheel, and EA the depth of the buckets; then, according to experiments on water-wheels, it appears that the rotatory form of the water in the buckets is nothing at c and d, and