164 General Analytical Theorems [OH. vu 



The last integral 



and this, as in the last theorem, is equal to 



_ 



dn dn J 



where S denotes summation over all conductors, including S. 

 This last sum of surface integrals vanishes, so that 



W - W = ^ (JJR* dxdydz through 8 



^-^-) 2 +--}^^^ through the field after 

 S has been introduced. 

 Thus W W is essentially positive, which proves the theorem. 



On putting the new conductor to the earth, it follows from the preceding 

 theorem that the energy is still further lessened. 



191. THEOREM. Any increase in the inductive capacity of the dielectric 

 between conductors lessens the energy of the field. 



Let the conductors of the field be supposed fixed in position and in- 

 sulated, so that their total charge remains unaltered. Let the inductive 

 capacity at any point change from K to K + SK, and as a consequence let 

 the potential change from V to V + SV, and the total energy of the field 

 from W to W +8TT. 



If E lt E%, ... denote the total charges of the conductors, V lt F 2 , ... their 

 potentials, and p the volume density at any point, 



xdydz, 

 so that, since the E's and p remain unaltered by changes in K, we have 



SW = $2ESV+jjjpSVda;dyd2 ............... (106). 



We also have 



so that 



= !IM(^) + C) + (^}\*Kdxdydz 



+ f- 1 1 1 K iV- ^F + ^ ^~ + ^ ^ I ^^^.-.(107). 



A^ / 1 ^/^ g/p ^ O y fig fa ( 



