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and Maclaurin, which Mr Smeaton has censured, as leading to conclusions altogether wide of the truth.
Let c be the velocity of the stream, v that of the wheel, A the area of the part of the float-board immersed in the water, g the velocity which a heavy body acquires in one second when falling freely.' Then c-u will be the relative velocity of the stream and the wheel, or the velocity with which the water strikes the wheel; and if we take h, a fourth proportional to go, (C-5): and 1 g, h will be the height from which a body must fall to acquire the velocity c-v, and will be equal to ul. Wherefore, by a prop. well known in Hydraulics, the circumference of the wheel is urged by the weight of a column of water, of which the section is A, and the height ! - and of which the solidity is therefore Axle ). Thus far the investigation is
2g applicable to all undershot wheels, and to all hydraulic engines of a similar construction. But to bring it to the case of those experiments in which the expenditure of the water was the same, let E be equal to that expenditure, that is, to the cubic inches delivered in one second, then A==, and so the pressure of the water, or the intensity of the impelling power = (-u).
Now, if the resistance overcome, or the weight raised, be = W, and its velocity u, then when the machine has attained a state of uniform motion, the momenta of the resisting and impelling forces must be equal; that is, W xu= (-v)'v. The quantity W x u will therefore be a maximum, when (
c v)?w is a maximum, the coefficient being constant, and v alone being variable. But (
c v)*v is a maximum, as is easily shown, when v= c, and therefore W xu is also a maximum in that case. Now, W X u has already been shown to be the measure of the effect of the machine; therefore, the effect of the machine is greatest when v= 4, that is, when the velocity of the wheel is one third of the velocity of the water which impels it. In that case, also, the quantity (
c v)v, which is always equal to the effect. becomes E 46 C-E , 4c.
X27; therefore the maximum of effect is proportional to ca or to the square of the
velocity of the stream ; which is precisely Mr Smeaton's third maxim, as deduced from his own experiments. In the same way might the truth of his fourth maxim, that, when the aperture of the sluice is given, and when the height of the head varies, the effect is as the cube of the velocity of the water, be deduced from the received principles of Hydraulics ; and yet Mr Smeaton evidently considers both these maxims as inconsistent with those principles. Here, therefore, this ingenious man has evidently fallen into an error; and this he has done, either from not having thoroughly considered the measure which he himself had assigned to the effect, or from not clearly perceiving that when a machine comes to its state of uniform motion, it is not because the resisting and impelling powers are equal, but because their momenta are equal; that is to say, the resisting power multiplied into its velocity, equal to the impelling power multiplied into its velocity.
We have gone over the above investigation with more minuteness than a matter of so little difficulty may seem to require ; because we wished that it might appear quite plain from what sources all the reasoning was derived, and that no part of it involved the idea of mechanical power, or of the vis viva. The only part of it that can be suspected of doing so, is the hydraulic proposition concerning the impulse of a Auid; but it is certain that this theorem, though one of those about which some difference of opinion has arisen, can be deduced in a satisfactory manner from the nature of fluids, without any appeal to the doctrine of the vis viva.
At the same time that the theorems, derived in the manner exemplified above, bring out results that agree with Mr Smeaton's experiments as to the proportion between effects arising from different impulses, yet it must be acknowledged that they do not agree well as to the absolute quantities. Thus, the greatest effect is assigned by the theory, in the case of undershot wheels, to that velocity of the wheel which is one third of the velocity of the stream. Mr Smeaton found, that the velocity which gives the greatest effect, is between and , but nearer to the latter.
The theory gives the maximum load of the impelling power ; Mr Smeaton's experiments make these quantities approach very near to equality. Whatever be the reason of a difference that falls out on the opposite side to what is usual, the practical result being more favourable than the theoretical, it certainly does not lye in any thing which the introduction of the vis viva would correct. Were we, for example, to suppose that the forces of the single particles of the fluid, to put the wheel in motion, are as the squares of the velocities, we should then have, for the expression of the impelling force, ( c v)s multiplied into some con
squares of ches of the fluidmple, to s
body has moved while that force acted on it; and the sum of all these products will be proportional to the square of the velocity, and, of course, the square root of the said sum, to the velocity itself.
Now it is obvious, that in order to apply this theorem to any case, we must be able to express the forces in terms of the distances at which they act; for then the sum of the products described in the theorem will either be found by the summation of series, or the quadrature of curves; so that the thing wanted will be determined. The circumstance, therefore, which distinguishes the one of these kinds of dynamical problems from the other, is, whether the forces that produce the motion can be most easily expressed in terms of the time reckoned from a given instant, or in terms of the distance reckoned from a given point. Instances of both cases are easy to be given. Suppose it required to determine the velocity of a body accelerated or retarded by the action of a constant force, as heavy bodies are in their descent or ascent at the surface of the earth; In this case, either of the two methods may be employed indifferently. The force being given, if it be multiplied into the time during which it acts, the product will be proportional to the velocity, according to the first proposition. And in the same way, if the given force be multiplied into the distance passed over, the square root of the product will be proportional to the velocity; and thus, in either way, may the velocity, with nearly equal facility, be determined. It must be determined in both ways to make the investigation complete ; and it is a matter of indifference with which we begin.
But it is not so if the accelerating force is variable, and ex. pressed by some function of the distance from a given point, (as gravitation really is when we take in a considerable range): the first step in the inquiry must be made by help of the second proposition, that is, by multiplying the force into the fluxion of the distance from the said point, and making the Auent (which will easily be found) equal to the square of the velocity. The velocity being thus expressed in terms of the distance, the time required for moving over a given distance will next be found. It is in this way that Newton has resolved the very problem here proposed, in the 39th proposition of the first book of the Principia, before referred to. It is therefore according as the data in any problem furnish means for integrating one or other of the formulas derived from the propositions above mentioned, that the one or the other must be employed in the solution of that problem. In the use of this second method, however, there is a circum-.