290 ELECTROSTATICS. [PT. II. CH. VII. 



151. Green's Function. As a means of solving certain 

 problems in electrostatics Green introduced a certain function*, 

 whose properties we shall next consider. Green's problem for a 

 portion of space T bounded by a closed surface S may be stated as 

 follows : 



It is required to find a function G satisfying the following 

 conditions : 



1. G is harmonic in the whole space considered with the ex- 

 ception of a single point P. 



2. G becomes infinite at P, but in such a manner that the 



function G -- is harmonic, r being the distance from the pole P. 



3. The value of any function V harmonic in r is given at the 

 pole P by the surface integral 



* ff 



4>7rJJ s 



A function satisfying these conditions is called Green's function 

 for the space r and pole P. 



The problem is unique, if it has a solution. For if there are 

 two solutions GI and 6r 2 , by 3, 



so that by subtraction 



for any harmonic function V. But by 2, 



^_1 an d G 2 -- 

 r r 



are harmonic, so that their difference GI 6r 2 is also harmonic. 

 Applying the above result to the harmonic function G l - G z , 



(s) 



* Green, Essay, 5. The name Green's Function is due to C. Neumann, who 

 applies it, however, as does Maxwell, to the function G - - . 



