254 ELECTKOSTATICS. [PT. II. CH. VI. 



due to the fixed electrification of D in which the volume density is 

 p. The total energy of the system will be 



(i) W = 



or 



vy 



(2 > r 'i//J. 



3s 



Suppose now that if we change p in the conductors to 

 and o- to <r + So-, the integral W becomes W + S W, while p and V 

 are unchanged at all points, since the electrification of D is un- 

 changed. Also since the charges of the conductors are to be un- 

 changed, we must have for any conductor K s 



(3) Se s = 8 pdr + ^ adS = 0. 



The condition for stable equilibrium is that for all possible 

 values of the functions &p and So- subject to the conditions Se s = 0, 

 we must have SW > 0, ( 58). 



Making the above changes in the integral (i), we have 

 (4) 



and subtracting W, we get 



