97-99] MOTION OF TWO SPHERES. 143 



which, in the neighbourhood of A becomes equal to 



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



a G b 3 cos 6 



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

 of A, 



and at the surface of /?, 



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

 kinetic energy, viz. 



(i), 



we have P=-p Jj ^dS A = 'jppa? (l+% -^ J , 



.. (ii). 



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 



(iii). 



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, " On the Motion of Two Spheres in 

 a Fluid," Phil. Trans., 1880, p. 455; K. A. Herman, "Cn the Motion of Two 

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

 Hydrodynamische Untersuchungen, Leipzig, 1883. 



