THEOREMS RELATING TO CLOSED SURFACES. 43 



V 



the outside is. therefore between V and zero. It cannot be equal 

 to V, unless V itself is zero. In fact, it cannot be equal to V 

 without being a maximum in reference to adjacent points, or without 

 forming part of a space at constant potential which would extend 

 to the surface S ; but, from the preceding theorem, the potential 

 in this latter case would be constant throughout the dielectric, and 

 could only have the value zero, for it is zero at an infinite distance. 



56. A conducting surface S, which contains all the acting masses, 

 can only have electricity of one kind. 



For let V be the potential of the surface, which we will suppose 

 is positive. If there were negative electricity at a point A, lines 

 of force would terminate there, and there must be somewhere an 

 external point where the potential has a higher value than V, which 

 is impossible from the preceding theorem. 



57. When, in a system in equilibrium, a conductor envelopes 

 diverse masses of electricity, the algebraical sum of the quantities of 



Fig. 13- 



electricity in the interior, and on the internal surface of the body, 

 is zero. 



Let m, m ', m" ... be the masses comprised within the interior 

 of the conductor A (Fig. 13), and M the mass of the layer spread 

 on the internal surface S ; to which we may add that there may 

 be electricity on the external surface S', and other masses beyond. 

 On a closed surface S 1} in the conductor, and comprising all the 

 internal masses, the force is null at each point 



The flow of force relative to this surface is therefore null, and 

 accordingly the algebraical sum of the masses which it contains is 

 null. We have then 



M + m + m' + m" + ..... = 0, 

 whence 



