179, 180] DIELECTRICS AND MAGNETIZABLE BODIES. 355 



the maximum theorem of 119, which we shall also assume to 

 hold.* We have now 



(4) W = ' VdS + P Vd 



Letting V vary without changing the charges or, p, 

 (5) W + & r W=<r(V + SV)dS+ p(V+SV)dr 



to dy dz 



from which 



+ 



(6) BrW=jj<r$rdS+jjj 



9F9SF 9F9SF 9F9SF) . 

 1- -- -- h - dr 



1 fff (9F9SI 



III AM 3 ~cT~ 



i fff f/sSFy , /8SFv /asFvi , 



.. -^rllL' J '\(^) + (w) + (^)\ dT ' 

 and by Green's theorem applied to the third integral 



fff r i (a / ar\ 9 / 9F\ , 9 / a 



p + T" 15- (P-TT +5T (A*-OT r+r IM-O" 

 jjJL- 4>7r l^ ^ 9a? ^ 9 2/ v ?y / 9^ v dz 



If now the energy of the actual distribution of potential is 

 to be a maximum for all possible values of 8F, the first two in- 

 tegrals must vanish, which can be the case only if throughout 

 space we have 



1 9 / 9F\ 9 / 9F\ 9 / 9F 

 ? = - 



* See Helmholtz, Wiss. Abh. Bd. i., p. 805. 



232 



