180 



(2-2) 



CO. 



DR. T. J. FA. BROMWICH ON THE 



a 2 v K 3 a v 



*"\ 9 2 **\x2 



f4-t** /* f\T 



Or C Vb 



1 8 2 V 1 _9_ /K 3U\ 

 r 30 3?' p 30 \ c St / ' 



cy = - 



1 3 2 V 1 a /K3U 



p 30 3r r 30 \ c 3 ; 

 where U, V are any two solutions of the equation 



(2-3) 



3 2 U 



3r a r- sin 





au 



i 8 2 u 



A solution of (2 '3) which is sufficiently general for the applications in view may be 

 found by assuming that U and V can be expressed as sums of terms of the type 



It is easy to see that then (2'3) leads to the equation 



(2-4) 



Ylsin030\ 30 / ' sin 2 



3 2 Y 



and since the two sides of equation (2'4) are functions of r, t and of 6, <j> 

 respectively, it is clear that each side must be a mere constant. If we write the 

 constant in the form n(n+l), it is evident that Y must be a surface-harmonic of 

 order n. 



Accordingly in problems (such as those with which we shall be concerned in the 

 sequel) where the whole of angular space is considered, the value of n must be a 

 positive integer ; for (except when n is an integer) there are no surface-harmonics 

 which are everywhere continuous and single- valued. 



Thus we may reduce our solution to the form 



(2-5) U or V = 2F B (r, OY.(0, *), n = 0, 1, 2, 3, ..., 



where F w is a solution of the equation 



(2-6) 

 and 



8 2 F, n(n+l) v 

 8* 2 r 2 



c, 2 = 



