208 Methods for the Solution of Special Problems [OH. vm 



distance / from its centre 0, and let Q be any point on the surface of 

 the sphere. 



Let PQ be equal to R, so that 



R* = f*+a 2 - 2af cos POQ. 

 We have the identity 



-* 



where the integration is taken over the surface of the sphere, a result 

 which it is easy to prove by integration. 



e 

 A point charge e placed at P induces surface density -- 



on the surface of 



ea 



the sphere ( 214), and the total induced charge is - - ? -. The identity is therefore 

 obvious from electrostatic principles. 



FIG. 72. 



Now introduce a quantity u defined by 



.(141), 



so that u is a function of the position of P. If P is very close to the 

 sphere, f z a 2 is small, and the important contributions to the integral arise 

 from those terms for which R is very small : i.e. from elements near to P. 



If the value of F does not change abruptly near to the point P } or 

 oscillate with infinite frequency, we can suppose that as P approaches the 

 sphere, all elements on the sphere from which the contribution to the 

 integral (141) are of importance, have the same F. This value of F will of 

 course be the value at the point at which P ultimately touches the sphere, 

 say F P . Thus in the limit we have 



'*-"*// (142) , 



u 





= FP -~ , by equation (140) 



when in the limit / becomes equal to a. 



