148 Dr. T. J. I'a. Bromwich on 



We can now write out the complete solution of the first type 

 in the form 



x = 



Y = 



Z = 



ap 



1 B 2 U 



CBf\c a*/' 



>(l-6) 



BaW 



1 B 2 U __1 A^^il\ 



cafa? C7 ~ B-d v \c a*> j 



where U is an arbitrary solution of equation (1*5). 



Similarly, by starting from X = 0, we obtain the complete 

 solution of the second kind : 



x= o, 

 ~ carv a*/' 



z_ + Ba^ a*)' 



c« = 



a 2 v 

 ' ap 



c/3 = 



i a 2 v 

 "Ba^av 





i a 2 v 



a* 2 ' 



(1-7) 



67 "c ajar j 



here V is another arbitrary solution of equation (1*5). 



§ 2. Discussion of the solution of § 1. 



In order to determine conditions under which we can 

 assert that the most general solution of equations (E)' 

 and (M) can be deduced by superposing the two solutions 

 (1*6) and (1'7), it will be necessary to show that all the 

 other components of force must vanish when X = and 

 a — 0. For then it is evident that all the components are 

 uniquely determined by means of the two X and a. 



The proof given below refers specially to spherical polar 

 coordinates, and to problems in which the whole of angular 

 space is used. The earlier part of the proof is arranged 

 so that it can be applied to other types of orthogonal 

 coordinates ; but the details of the final reasoning need 

 modification, and must be adapted in other cases so as to 

 suit the problem in hand*. 



If we begin by writing X=0, a = 0, it will be evident that the 

 preliminary analysis used in § 1 is not affected: and we can 

 express our conclusions in the same form (1*6), except that now 



* It may be useful to remind readers that the conclusion that all the 

 other components of force must vanish when X = 0, and a — is not 

 necessarily true for all types of orthogonal coordinates ; an example 

 to 'the contrary is given by Lord Kayleigh's solution found in §7 

 below. 



