162 General Analytical Theorems [OH. vn 



The proof is almost identical with that of the last theorem, the only 

 difference being that at every point of the surfaces, we have 



instead of the former condition V F=0. We still have 



so that equation (105) is true, and the result follows as before, except that 

 V and V may now differ by a constant. 



188. Theorems exactly similar to these last two theorems are easily 

 seen to be true when the dielectric is different from air. 



For, let V, V be two solutions, such that 



I \ K I- (V- F')j + ~ \K I (V- F')l + j- \K I (V- V')\ = 



c>x ( &e v '] dy ( dy '} dz { dz^ '} 



at all points of the space, and at the surface either F V = 0, or 



A<7-FO-a 



By Green's Theorem, 



= by hypothesis. 

 Equation (105) now follows as before, so that the result is proved. 



COMPARISONS OF DIFFERENT FIELDS. 



189. THEOREM. If any number of surfaces are fixed in position, and a 

 given charge is placed on each surface, then the energy is a minimum when 

 the charges are placed so that every surface is an equipotential. 



Let V be the actual potential at any point of the field, and V 

 the potential when the electricity is arranged so that each surface is 



