131] ELECTRICAL EQUILIBRIUM. 255 



Now F, F, SV are potentials due respectively to distributions of 

 densities p, a- in the space K for F, p in the space D for V, and 

 S/o, &r in the space K for SF, and accordingly by Gauss's theorem 

 of mutual potential energy, 117 (5), 



(6) jj <rSVdS=jj VSrdS, 



In virtue of these equalities, the integral reduces to 



all the integrals being taken throughout all the conductors only. 

 In order to take account of the conditions $e s = we must multi- 

 ply each such equation by an arbitrary constant, c s , and add the 

 sum to the above value of SW, 



(8) 8TF-S,c.&.0, 



that is, = to the first order of small quantities, while the terms 



of second order must be positive for a minimum. 



Introducing the values of Be, (3) 



pdr + BJj vdS^jjf Spdr+jj 



we get 



(10) 2 S jjJJ : (F+ V - c s ) tpdr+jj^ (F+ V - c s ) 



+ I [[( SVSpdr+ ft SVSo-dsl ^ 0. 



^ J JJ K g J J K 8 ), 



The equations of condition having been introduced, we may 

 treat Sp and So- as arbitrary, and if we put in each conductor 



(n) F+F'-c, = 0, 



the above reduces to the terms of second order 



