240] Spherical Harmonics 209 



If the value of F oscillates with infinite frequency near to the point P, we obviously 

 may not take F outside the sign of integration in passing from equation (141) to 

 equation (142). 



If the value of F is discontinuous at the point P of the sphere with which P 

 ultimately coincides, we again cannot take F outside the sign of integration. Suppose, 

 however, that we take coordinates p, $ to express the position of a point P' on the surface 

 of the sphere very near to P', the coordinate p being the distance PP' t and S being the 

 angle which PP' makes with any line through P in the tangent plane at P. Then F 

 may be regarded as a function of p, $, and the fact that F is discontinuous at P is expressed 

 by saying that as we approach the limit p=0, the limiting value of F (assuming such a 

 limit to exist) is a function of 3 i.e. depends on the path by which P is approached. 

 Let F (5) denote this limit. Then 



= / 2 _z^ 2 



= J- (F($) (-)) flW, by equation (140). 

 Z7r J \JJ 



On passing to the limit and putting =/, we find that 



u- JLjX*)** .................................... (143), 



i.e. u is the average value of F taken on a small circle of infinitesimal radius surrounding 

 P. In particular, if F changes abruptly on crossing a certain line through P', having a 

 value FI on one side, and a value F 2 on the other, then the limiting value of u is 



If we take 6 to denote the angle POQ, 



i = (/>-2a/c 



1 / a 2 

 ~7V 



If a 2 - 2a/cos d /a 2 - 2a/cos 0\ 2 1 



~7L *"" "7^ Mrx^ 7 ~) ' "T 



or, arranging in descending powers of f t 



a 2 n a s 



- 2 +P 33 +... ............ (144), 



in which P 1? P 2 , P 3 ... are functions of ^, being obviously rational integral 

 functions of cos 0. When 6 0, 



W "/' 



J. 



