PROBLEMS IN THREE DIMENSIONS. 



[CHAP. V 



satisfied at the surface of the sphere A, and in order to neutralize the normal 

 velocity at this surface, due to H lt we must superpose a double-source at 7/ 2 , 

 the image of ff 1 in the sphere A. This will introduce a normal velocity at the 

 surface of B, which may again be neutralized by adding the image of H 2 in B, 

 and so on. If /ij, ^ 2 , /* 3 , ... be the strengths of the successive images, and 

 /u / 2 ,/ 3 , ... their distances from A, we have, if AB=c, 



(v), 



and so on, the law of formation being now obvious. The images continually 

 diminish in intensity, and this very rapidly if the radius of either sphere is 

 small compared with the shortest distance between the two surfaces. 



The formula for the kinetic energy is 



provided 



d<f> 



dn 



dS K 



(vii), 



where the suffixes indicate over which sphere the integration is to be effected. 

 The equality of the two forms of M follows from Green's Theorem (Art. 44.) 



The value of near the surface of A can be written down at once from the 

 results (6) and (7) of Art. 86, viz. we have 



^ + . . . V cos B + &c.. . . 



the remaining terms, involving zonal harmonics of higher orders, being omitted, 

 as they will disappear in the subsequent surface-integration, in virtue of the 

 conjugate property of Art. 88. Hence, putting d<J>/dn= cos#, we find with 

 the help of (v) 



It appears that the inertia of the sphere A is in all cases increased by the 

 presence of a fixed sphere B. Compare Art. 92. 



