ducing the condition (44), we have, omitting the constant, or supposing A,,,= 0,

0,0

Determining the coefficients such as Am, from the condition (45) in the usual manner we have, m and n being > 0,

4c

рп

a

Am, n= — (e *P - e-p*) - 1 [ " [f(x,y) cosmræ/a.cosnry/b.dxdy,

Ao, n

πραι

πρ

е

0

— e-nxob) - 1 [ " ["ƒ (x, y) cos nπy/b. dxdy*,

(enπc/b e-nπc/b)

with a similar expression for Am, whence the value of o corresponding to f (x, y) is known. In a similar manner we may find the values corresponding to the similar functions belonging to each of the other faces. If W' be the quantity corresponding to W for the face opposite to the plane xy, and U, U', correspond to W, W', for the faces perpendicular to the axis of x, and if V, V', be the corresponding quantities for y, there remains only to be found the part of due to these six quantities. Since U, U', are the velocities parallel to the axis of a of the faces perpendicular to that axis, and so for V, V', &c., the motion corresponding to these six quantities may be resolved into three motions of translation parallel to the three axes, the velocities being U, V and W, and that motion which is due to the motions of the faces opposite to the planes yz, zx, xy, moving with velocities U'-U, V'- V, W'-W, parallel to the axes of x, y, z, respectively. The condition that the volume of the fluid remains the same requires that

satisfy all the requisite conditions. Hence the part of due to

* The function f(x, y) in these integrals may be replaced by F(x, y), since

the six quantities U, U', V, V', W, W', is

x2 2a

y2

Ux + Vy + Wz + (U' − U ) 2 2 + (V' - V ) 2 + (W' - W ) 2.

26

(W'_W)

This quantity, added to the six others which have already been given, gives the value of which contains the complete solution of the problem.

The case of motion which has just been given seems at first sight to be an imaginary one, capable of no practical application. It may however be applied to the determination of the small motion of a ball pendulum oscillating in a case in the form of a rectangular parallelepiped, the dimensions of the case being great compared with the radius of the ball. For this purpose it will be necessary to calculate the motion of the ball reflected from the case, by means of the formulæ just given, and then the motion again reflected from the sphere, exactly as has been done in the case of a rigid plane, Art. 10. In the present instance however the result contains definite integrals, the numerical calculation of which would be very troublesome.

[From the Cambridge Mathematical Journal, Vol. IV. p. 28. (Nov. 1843).] ON THE MOTION OF A PISTON AND OF THE AIR IN A CYLINDER.

WHEN a piston is in motion in a cylinder which also contains air, if the motion of the piston be not very rapid, so that its velocity is inconsiderable compared with the velocity of propagation of sound, the motions of the air may be divided into two classes, the one consisting of those which depend directly on the motion of the piston, the other, of those which are propagated with the velocity of sound, and depend on the initial state of the air, or on a breach of continuity in the motion of the piston. If we suppose the initial velocity and condensation of the air in each section of the cylinder to be given, and also the initial velocity of the piston, both kinds of motion will in general take place, and the solution of the problem will be complicated. If, however, we restrict ourselves to motions of the first class, the approximate solution, though rather long, will be simple. In this case we must suppose the inital velocity and condensation of the air not to be given arbitrarily, but to be connected, according to a certain law which is yet to be found, with the motion of the piston. The problem as so simplified may perhaps be of some interest, as affording an example of the application of the partial differential equations of fluid motion, without requiring the employment of that kind of analysis which is necessary in most questions of that sort. It is, moreover, that motion of the air which it is proposed to consider, which principally affects the motion of the piston.

Conceive an air-tight piston to move in a cylinder which is closed at one end, and contains a mass of air between the closed end and the piston. For more simplicity, suppose the rest of the

1

cylinder to contain no air. Let a point in the closed end be taken for origin, and let x be measured along the cylinder. Let be the abscissa of the piston; a the initial value of x1; u the velocity parallel to x of any particle of air whose abscissa is x; p the pressure, p the density about that particle; II the initial mean pressure; P1 the value of p when x=x x=x1; X, a function of x, the accelerating force acting on the air; then for the motion of the air we have

1

Now, k being very large, for a first approximation let

neglected; then, integrating (5),

be

k

Substituting in (4) the value of p when t=0, we have

so that p' and u' are small quantities of the order 1/k; then, substituting these values in (5) and (6), remembering that the quantities which are not small must destroy each other, and retaining only small quantities of the first order, we have

and the conditions (2), (3) and (4) give

u'=0 when x=0, or x=x, and t is positive...(9);

Substituting the values of p' and of its differential coefficients in (8), and integrating, we obtain