264 ELECTROSTATICS. [PT. II. CH. VI. 



I. To find for each value of s from 1 to n, a function^, which 

 at the surface K s takes the constant value 1, at all the surfaces^ 

 where r is different from s, takes the constant value 0, and in 

 all space external to K is harmonic. Each of these n problems 

 is a different problem of Dirichlet. 



II. To determine a function w, which in all the conductors is 

 zero, in the bodies D satisfies the equation 



Aw = 4<7rp, 

 and in the rest of space is harmonic. 



These n + 1 functions being found, the required function V is 

 given by the linear function 



(i) V= V&+ V 2 v 2 ...+ V n v n + w, 



where V lt F 2 ... V n are the given constant values. For each of 

 the functions v s and w is harmonic in all space except D, where 

 the v 8 'a are harmonic, and w satisfies Aw = 4-7T/3; therefore the 

 sum V is harmonic everywhere except in D, where it satisfies 



On any conductor K s , w and all the vs vanish except v s , which 

 is 1, hence 



V ' V 



- " s- 



From any of the functions v s and w let us calculate for any 

 surface K r the integrals 



1 [f dv S jv n l [f dw JQ 



q sr = - 6 dS, Qr = --j~ ^-dS. 

 47rJJ Kr dn e 4<7rJJ Kr dn e 



Since the finding of the function v s is a purely geometrical 

 problem, depending on the form and position of the surfaces K S) 

 all the ri* quantities q rs are geometrical constants for the given 

 system of conductors. We have now for the charge of any con- 

 ductor K s 



= _1 ff 



4>7rJJ 



4-rr 



or inserting the above notation for the integrals, 

 (3) e s = q^ 4- q 2s V 2 . . . + q ns V n + Q.. 



