136 138] SYSTEMS OF CONDUCTORS. 267 



The coefficients p are called coefficients of potential Their 

 dimensions are = \ -f- \ Since the determinant A is un- 



LJ LAJ 



changed by interchanging columns and rows, the determinant 

 of the p's must have the same property, or p rs = p 8r . We may 

 prove this directly as we did for the q's. Let V be the value of 

 the potential when K r has the charge e and all other conductors 

 charge 0. Let V be the value of the potential when K s has 

 charge e and the others charge 0. Then as in (5) 



and since on any conductor Ki, V and V ' ' are constant and re- 

 spectively equal to V iy V{ ', 



Now since the potential V is due to a distribution in which only 

 K 8 is charged, all the integrals 



^n C 



vanish except for i = s, for which the integral is 4?re, likewise all 

 the integrals 



vanish except for i = r, when the value is 4tire. Consequently 

 we have 



(13) 



Now from the equations (10), putting e r = e, the other es zero, 



V s =p rs e. 

 Again putting e s = e, the others zero, 



V r '=p sr e. 

 Whence 



04) Prs=Psr, 



and we have the reciprocal theorem : 



If a conductor A receive a certain charge, e, all the other con- 

 ductors of the system being uncharged, the potential of any other 

 conductor B is the same as would be attained by A if E should 

 receive the charge e, all the other conductors being uncharged. 



