XVI. On certain Conditions under which a Perpetual

Motion is possible.

BY GEORGE BIDDELL AIRY, M.A.

MEMBER OF THE ASTRONOMICAL SOCIETY, FELLOW OF TRINITY COLLEGE, AND OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY, AND PLUMIAN PROFESSOR OF ASTRONOMY

AND EXPERIMENTAL PHILOSOPHY, IN THE UNIVERSITY

OF CAMBRIDGE.

[Read Dec. 14, 1829.]

It is well known that perpetual motion is not possible with any laws of force with which we are acquainted. The impossibility depends on the integrability per se of the expression Xdx + Ydy + Zdz: and as in all the forces of which we have an accurate knowledge this expression is a complete differential, it follows that perpetual motion is incompatible with those forces.

But it is here supposed that, the law of the force being given, the magnitude of the force acting at any instant depends on the position, at that instant, of the body on which it acts. If however the magnitude of the force should depend not on the position of the body at the instant of the force's action, but on its position at some time preceding that action, the theorem that we have stated would no longer be true. It might happen that, every time that the body returned to the same position, its velocity would be less than at the preceding time: in this case the body's motion would ultimately be destroyed. On the contrary it might happen that the body's velocity in any position

would be more rapid every time than at the time previous. In this case the velocity would go on perpetually increasing: or the velocity might be made uniform if the machine were retarded by some constantly acting resistance: or, in other words, the machine might move with uniform velocity, and might at the same time do work: which is commonly understood to be the meaning of the term perpetual motion. If the machine had no work to do, the increasing friction, &c. would operate as an increasing work, and the velocity would be accelerated till the acceleration caused by the forces was equal to the retardation caused by the friction: after which it would remain unaltered.

For this idea I am indebted to the admirable account of the organs of voice given by Mr. Willis. The phenomenon to be explained was this. When two plates are inclined at an angle greater than a certain angle, it is found that the effect of a current of air passing between them is to give a tendency to open wider. When they are inclined at any angle smaller than that certain angle, the effect of the current is to make them collapse. If then the plates be supposed to vibrate through the position corresponding to that angle, the tendency of the forces at all times is to bring them to that position. Each plate is in the state of a vibrating pendulum and whatever be the law of force which acts on it it is certain that if the force be the same when the plate is in the same position, this force will have no tendency to increase the velocity. The retardation arising from friction, &c. will therefore soon destroy the motion. But it is found, in fact, that the motion is not destroyed. What then is the accelerating force which keeps up the motion? Mr. Willis explains this by supposing that time is necessary for the air to assume the state and exert the force corresponding to any position of the plate which is nearly the same as saying that the force depends on the position of the plate at some previous time. In this

Paper, which is intended to investigate the mathematical consequences of an assumed law, I shall not discuss the identity of these suppositions: I shall only remark that the general explanation appears to be correct, and that it clears up several points which always appeared to be in great obscurity.

Let us now consider the case of a vibrating body acted on by two forces, of which one is proportional to its actual distance from the point of rest, and the other proportional to its distance at some previous time. Putting (t) for the body's distance, the equation is

This equation I am unable to solve rigorously: but on the supposition that g is small, an approximate solution may be obtained from the formulæ in the Memoir on the Disturbances of Pendulums, &c. (Cam. Trans. Vol. III. p. 109.) Neglecting at first the small term, we have

and therefore ƒ in the formulæ alluded to is

=ag.sin (t√e + b − c √e).

The increase therefore of the arc of semi-vibration is

Je S.ag, sin (t √e + b − c √e) cos († √ē + b)

To find the increase from one vibration to another we must

take the integral between two values of t differing by

thus we obtain for the increase

.ag π

sin c√e

e

I shall not occupy the time of the Society by a discussion of the different values of the increase corresponding to different values of c: I shall only remark that if ce be less than T, the arc of vibration increases continually. Nor shall I consider the cases in which c is supposed to be a function of the position or velocity of the vibrating body (which possibly might better represent the circumstances that originally suggested this investigation.) My object is gained if I have called the attention of the Society to a law hitherto (I believe) unnoticed, but not unfruitful in practical applications.

G. B. AIRY.

OBSERVATORY, Dec. 13, 1829.

XVII. Some Observations on the Habits and Character

of the Natter-Jack of Pennant, with a List of Reptiles found in Cambridgeshire.

BY THE REV. LEONARD JENYNS, M.A. F.L.S.

AND FELLOW OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY.

[Read Feb. 22, 1830.]

IT is observed by Dr. Fleming, in his British Animals, that "the history of the Natter-Jack, like that of many of our native reptiles, is involved in obscurity." Under the sanction of such a remark from one of our first Naturalists, I am induced to offer a few observations which I have had an opportunity of making upon the habits of this species, and to record it as a native of Cambridgeshire.

This animal does not appear to have been often noticed in this country. Its first discoverer, I believe, was the late Sir Joseph Banks, who found it near Revesby Abby in Lincolnshire, and sent an account of it to Pennant, stating that it was known in the above district by the name of Natter-Jack. This account was afterwards published by Pennant in his British Zoology, (Vol. III. p. 19.) who also says that it occurs on Putney Common.-I am not aware that it has been mentioned by any other British writer, otherwise than with reference to Pennant's original description.