4l0 Mb. STOKES'S SUPPLEMENT TO A MEMOIR 



be fixed relatively to the vessel, and let the axis of x be parallel to the generating lines of tlie cylin- 

 drical surface. The instantaneous motion of the vessel may be decomposed into a motion of transla- 

 tion, and two motions of rotation about the axes of y and z respectively; for by hypothesis there is 

 no motion of rotation about the axis of .v. According to the principles of my former papei', the 

 instantaneous motion of the fluid will be the same as if it had been produced directly by impact, the 

 impact being such as to give the vessel the velocity which it has at the instant considered. We may 

 also consider separately the motion of translation of the vessel, and each of the motions of rotation ; 

 the actual motion of the fluid will be compounded of those which correspond to each of the separate 

 motions of the vessel. For my present purpose it will be sufficient to consider one of the motions of 

 rotation, that which takes place round the axis of z for instance. Let w be the angular velocity 

 about the axis of z, a> being considered positive when the vessel turns from the axis of x to that of 

 y. It is easy to see that the instantaneous motion of the cylindrical surface is such as not to alter 

 the volume of the interior of the vessel, supposing the plane ends fixed, and that the same is true of 

 the instantaneous motion of the ends. Consequently we may consider separately the motion of the 

 fluid due to the motion of the cylindrical surface, and to that of the ends. Let (p^ be the part of 

 (p due to the motion of the cylindrical surface, (p,. the part due to the motion of the ends. Then we 

 shall have 



<}>=<S>,+ <Pe (»)• 



Consider now the motion corresponding to a value of rf), way. It will be observed that wxy 

 satisfies the equation, ( (36) of my former paper,) which <p is to satisfy. Corresponding to this 

 value of (p we have 



u = wy, V = fti.r, w = 0. 



Hence the velocity, corresponding to this motion, of a particle of fluid in contact with the cylindrical 

 surface of the vessel, resolved in a direction perpendicular to the surface, is the same as the velocity 

 of the surface itself resolved in the same direction, and therefore the fluid does not penetrate into, 

 nor separate from the cylindrical surface. The velocity of a particle in contact with either of the 

 plane ends, resolved in a direction perpendicular to the surface, is equal and opposite to the velocity 

 of the surface itself resolved in the same direction. Hence we shall get the complete value of (p by 

 adding the part already found, namely wxy, to fwice the part due to the motion of the plane 

 ends. We iiave therefore, 



^ = wxy + 20, = 2^^ -toxy, by (1) (2), 



and (p^ - <p^= uxy (3). 



Hence whenever either (p^ or <p^ can be found, the complete solution of the problem will be 

 given by (2). And even when both these functions can be obtained independently, (2) will enable 

 us to dispense with the use of one of them, and (3) will give a relation between them. In this case 

 (3) will express a theorem in pure anah'sis, a tiieorem which will sometimes be very curious, since 

 the analytical expressions for <p^ and (p^ will generally be totally different in form. The problem 

 admits of solution in the case of a circular cylinder terminated by planes perpendicular to its axis, 

 and in the case of a rectangular parallelepiped. In the former case, the numerical calculation of the 

 moments of inertia of the solid by which the fluid may be replaced would probably be troublesome, 

 in the latter it is extremely easy. I proceed to consider this case in particular. 



Let the rectangular axes to which the fluid is referred coincide with three adjacent edges of the 

 parallelepiped, and let a, b, c be the lengths of the edges. The motion which it is proposed to cal- 



