THEOREMS RELATING TO CLOSED SURFACES. 41 



:_Si . 



there would still be a state of equilibrium, which, superposed on the 

 first, will give a new state of equilibrium, in which the total charge 

 will be zero on each of the conductors. 



In this case, from the preceding observation, the density must be 

 everywhere zero. We have, then, 



The distribution is therefore the same in both cases, and therefore 

 the equilibrium is singular. If the proposed system comprised fixed 

 masses, they might always be supposed to be upon infinitely small 

 conductors, and the reasoning would not be changed. The theorem 

 is therefore a general one. 



52. THEOREMS RELATING TO CLOSED SURFACES. If the potential 

 is constant on a dosed surface S, not containing any acting mass, it is 

 constant throughout the whole interior. 



The potential, in fact, could not vary in the interior of the surface 

 S without attaining at one point a maximum or minimum value, 

 which is impossible, as there is no electricity (46). 



If the surface in question is the external surface of a conductor, 

 it is seen that the potential is constant not only in the mass of the 

 conductor, which is a necessary consequence of the conditions of 

 equilibrium, but also in the cavities which this may contain. 



53. If a surface at constant potential comprises a portion of a 

 dielectric, the potential is constant not only in the interior of this 

 surface, but also in all the exterior space outside the acting masses. 



For, as the potential is constant in the interior, no line of force 

 can traverse the surface ; all those which could meet it are either 

 produced there or are absorbed which is impossible, since there is, 

 no electricity on the surface. Hence no line of force meets the 

 surface. 



As no line of force meets the surface, the potential is constan 

 on the outside close to it ; hence a new surface S' may be drawn 

 having the same property, to which the same reasoning may be 

 applied, and so on indefinitely, provided we always keep outside 

 the acting masses. 



54. This latter theorem was demonstrated by Gauss in a 

 different manner, by means of the following lemma : 



If a spherical surface contains no acting mass, the potential at the 



