292 ELECTROSTATICS. [PT. II. CH. VII. 



ducting and connected to earth. If o- G is the density of the 

 induced charge, 



1 



~ 



and (1) is 



Fp = -|l VtrcdS. 



Suppose that G is Green's function for a certain space, with 

 the pole P, whose co-ordinates are a, b, c, and that G' is Green's 

 function for the same space, but a different pole P' whose co- 

 ordinates are a', &', c. Then there exists the reciprocal relation 

 that the values of either function at the pole of the other are 

 equal. For 



' 



where the suffixes indicate from what point the distance is measured. 

 Now since F and F' are harmonic, by the property of the two 

 Green's functions G and G ', 



(8) 

 so that 



The last integral but one vanishes because F and F' are harmonic 

 functions, while on account of the surface values of F and F r , the 

 last becomes 



dn r p 



Since both the functions l/r> and !/?> are harmonic except at 

 their poles P and P', by constructing small spheres about the 

 points P and P' and proceeding as in 83, we find that the two 

 parts of the last integral destroy each other (each being equal to 



