210 Methods for the Solution of Special Problems [CH. vm 



and when = 7r, 



so that when = 0, 



P P 1 



-LI -L 2 ... 1, 



and when TT, 



-p > -p,=-p,....=i. 



It is clear, therefore, that the series (144) is convergent for = and 

 # = 7r, and a consideration of the geometrical interpretation of this series 

 will shew that it must be convergent for all intermediate values*. 



Differentiating equation (144) with respect to /, we get 



If we multiply this equation by 2/, and add corresponding sides to 

 equation (144), we obtain 



ET 



Multiplying this equation by , and integrating over the surface of the 



' 



sphere, we obtain 



or, by equation (141), 



i * r,* 



ds - 



If the function F is continuous and non-oscillatory at the point P, then 

 on passing to the limit and putting /= a, we obtain 



If F is discontinuous and non-oscillatory, then the value of the series on the right is 

 not f, but is the function denned in equation (143). 



Now it is known that 1/r is a spherical harmonic, so that we have 



where the differentiation is with respect to the coordinates of Q. Hence l/R 

 must be of the form (cf. 233) 



1 = 2(^ + ^)3, ........................ (147), 



* Being a power series in cos 6 it can only have a single radius of convergence, and this 

 cannot be between cos = 1 and cos0= -1. 



