296 ELECTROSTATICS. [PT. II. CH. VII. 



r and r being the distances of any point from A and B respec- 

 tively. Therefore, since I// is harmonic in the space containing 

 A, Green's function is for that space 



n l R l 

 (3) **r~ir" 



dG _ 1 dr R 1 dr' cos (^r) R cos 

 ^^ 2 + ~ rz ~~ ~~~ f 



(5) 



so that the density of an equipotential layer induced by a unit 

 charge at A on the sphere made conducting is 



1 [COS (UjT) R COS (&' 



~ ~~~ 



Now in the triangles OMB and MAO we have 

 a- = R 2 + r 2 - 2jRr cos 



- cos 



so that 



cos (np) _ E cos (mr f ) _ a 2 - (R 2 + r 2 ) R (&*_- (R* + r' 2 )) 



^ "^7 ^/2 9 P/*^ *" ^ 



which by ( i ) and (2) gives p , and 



The whole induced charge is 



47rJJ V ^ a r 2 



and if J. is an outside point by Gauss's theorem 



so that 



