97-99] MOTION OF TWO SPHERES. 143 



which, in the neighbourhood of A becomes equal to 



, 3 6 3 



J 6 -rcos&amp;lt;9, 



nearly. To rectify the normal velocity at the surface of A , we add the term 



T a 6 6 3 cos d 

 s -jT ~^- 



Stopping at this point, and collecting our results, we have, over the surface 

 of A, 



and at the surface of Z?, 



Hence if we denote by P, $, R the coefficients in the expression for the 

 kinetic energy, viz. 



r r -,7j. / .T7,s\ \ 



we have P 



/ J 



The case of a sphere moving parallel to a fixed plane boundary, at a 

 distance A, is obtained by putting b = a, v = v , c = 2A, and halving the conse 

 quent value of T ; thus 



This result, which was also given by Stokes, may be compared with that of 

 Art. 97 (xvii)*. 



99. Another interesting problem is to calculate the kinetic 

 energy of any given irrotational motion in a cyclic space bounded 

 by fixed walls, as disturbed by a solid sphere moving in any 

 manner, it being supposed that the radius of the sphere is small 



* For a fuller analytical treatment of the problem of the motion of two spheres 

 we refer to the following papers : W. M. Hicks, &quot; On the Motion of Two Spheres in 

 a Fluid,&quot; Phil. Trans., 1880, p. 455; E. A. Herman, &quot; Cn the Motion of Two 

 Spheres in Fluid,&quot; Quart. Journ. Math., t. xxii. (1887). See also C. Neumann, 

 Hydrodynamische Untersuchungen, Leipzig, 1883. 



