151] GREEN'S FUNCTION. 291 



But by Green's theorem this is equal to the volume integral 



which, as in 86 can vanish only if GI 6r 2 = const. That is, with 

 the exception of a constant, Green's function is unique. But as in 

 the employment of the function only its derivative is used, the 

 constant makes no difference. 



Since the function G is harmonic, we have by 33 (2) 



or transposing, 



cni r d' , 

 by 83 (6). If on the surface G = we obtain 



Consequently if we can solve Dirichlet's problem for the given space, 

 obtaining a harmonic function F which takes at the surface S the 

 values 



then the function G = F 4- - 



r 



solves Green's problem. Conversely if we can solve Green's problem 

 for the space and for any pole P, the equation (1) enables us to 

 find any harmonic function V from its values at the surface, 

 solving Dirichlet's problem. 



The problems of Green and Dirichlet are thus exactly 

 equivalent. 



In physical language, Green's function is the potential due to 

 a positive unit of electricity placed at the pole P together with 

 that of the charge which it induces on the surface 8 made con- 



192 



