TO THE THEORY OF ELECTRICITY. 40 



induce an electrical state in the shell ; then will this induced 

 state be such, that the total action on an electrified particle, 

 placed any where within it, will be absolutely null. 



For let V represent the value of the total potential function, 

 at any point p within the shell, then we shall have at its inner 

 surface, which is a closed one, 



(3 being the constant quantity, which expresses the value of the 

 potential function, within the substance of the shell, where the 

 electricity is, by the supposition, in equilibrium, in virtue of the 

 actions of the exterior bodies, combined with that arising from 

 the electricity induced in the shell itself. Moreover, V evidently 

 satisfies the equation = B F, and has no singular value within 

 the closed surface to which it belongs : it follows therefore, from 

 Art 5, that its general value is 



Y*& 



and as the forces acting upon p t are given by the differentials of 

 F, these forces are evidently all equal to zero. 



If, on the contrary, the electrified bodies are all within the 

 shell, and its exterior surface is put in communication with the 

 earth, it is equally easy to prove, that there will not be the 

 slightest action on any electrified point exterior to it ; but, the 

 action of the electricity induced on its inner surface, by the 

 electrified bodies within it, will exactly balance the direct action 

 of the bodies themselves. Or more generally : 



Suppose we have a hollow, and perfectly conducting shell, 

 bounded by any two closed surfaces, and a number of electrical 

 bodies are placed, some within and some without it, at will ; then, 

 if the inner surface and interior bodies be called the interior 

 system ; also, the outer surface and exterior bodies the exterior 

 system ; all the electrical phenomena of the interior system, 

 relative to attractions, repulsions, and densities, will be the same 

 as would take place if there were no exterior system, and the 

 inner surface were a perfect conductor, put in communication with 

 the earth ; and all those of the exterior system will be the same, 

 as if the interior one did not exist, and the outer surface were a 

 perfect conductor, containing a quantity of electricity, equal to 



4 



