132 PROBLEMS IN THREE DIMENSIONS. [CHAP. V 



92. The same method can be applied to find the motion 

 produced in a liquid contained between a solid sphere and a fixed 

 concentric spherical boundary, when the sphere is moving with 

 given velocity u. 



The centre of the sphere being taken as origin, it is evident, 

 since the space occupied by the fluid is limited both externally 

 and internally, that solid harmonics of both positive and negative 

 degrees are admissible; they are in fact required, in order to 

 satisfy the boundary conditions, which are 



d$/dr = u cos 0, 



for r = a, the radius of the sphere, and 



d&amp;lt;j&amp;gt;jdr = 0, 



for r = b, the radius of the external boundary, the axis of x being 

 as before in the direction of motion. 



We therefore assume 

 and the conditions in question give 



whence A = r . -u, B = J -u (2). 



t&amp;gt; 3 a 3 b B a 3 



The kinetic energy of the fluid motion is given by 



Wmt-ollt^dS, 



the integration extending over the inner spherical surface, since 

 at the outer we have d^jdr = 0. We thus obtain 



where m stands for f Trpa 3 , as before. It appears that the effective 

 addition to the inertia of the sphere is now 



* Stokes, 7. c. ante p. 130. 



