Mr stokes, on SOME CASES OF FLUID MOTION. 137 



and U, U', correspond to JF, 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 <p due to these six 

 quantities. Since U, U', are the velocities parallel to the axis of x 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 yx, xz, oey, moving with 

 velocities U' — U, V - V, W — W, parallel to the axes of a;, y, z, respectively. The condition 

 that the volume of the fluid remains the same requires that 



- (U"- U) + 7 (F'- F) + - (IF'- IF) = 0. 

 a c 



It will be found that the velocities 



„ = - (f7'_ U), v = l {V- V), w = - (IF'- IF), 

 a c 



satisfy all the requisite conditions. Hence the part of (p due to the si.\ quantities U, U', V, V, 

 W, IF', is 



Ua!+Vy+ Wz + ([/'- U) — + (V'-V)^ + (W- W) — . 



2a 20 2c 



This quantity, added to the six others which have already been given, gives the value of <p 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 formulas 

 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. 



G. G. STOKES. 



Pembroke College, 

 M»y, 1843. 



Vol.. VIII. Part I. 



