V; Electricity in Equilibrium. ;::'^^ |>1 



and further, it may I think be shown that either method, of 

 viewing the subject, when carried sufficiently far, may be made 

 the foundation of a mathematical theory which would lead to 

 the elementary principles of the other as consequences. This 

 theory would accordingly be the expression of the ultimate law 

 of the phsenomena, independently of any physical hypothesis we 

 might, from other circumstances, be led to adopt. That there 

 are necessarily two distinct elementary ways of viewing the 

 theory of electricity, may be seen from the following considera- 

 tions, founded on the principles developed in a previous paper 

 in this Journal*. 



Corresponding to every problem relative to the distribution 

 of electricity on conductors, or to forces of attraction and re- 

 pulsion exercised by electrified bodies, there is a problem in the 

 uniform motion of heat which presents the same analytical 

 conditions, and which, therefore, considered mathematically, is 

 the same problem. Thus, let a conductor A, charged with a 

 given quantity of electricity, be insulated in a hollow conducting 

 shell, B, which we may suppose to be uninsulated. According 

 to the mathematical theory, an equal quantity of electricity of 

 the contrary kind will be attracted to the interior surface of B 

 (or the surface of B, as we may call it to avoid circumlocution), 

 and the distribution of this charge, and of the charge on A, will 

 take place so that the resultant attraction at any point of each 

 surface may be in the direction of the normal. This condition 

 being satisfied, it will follow that there is no attraction on any 

 point within A, or without the surface of B, that is, on any 

 point within either of the conducting bodies. The most conve- 

 nient mathematical expression for the condition of equilibrium, 

 is that the potential at any point Pf must have a constant value 

 when P is on the surface of A, and the value nothing when P 

 is on the surface of B ; and it will follow from this that the 

 potential will have the same constant value for any point within 

 A, and will be equal to nothing for any point without the sur- 

 face of B. 



If A be subject to the influence of any uninsulated conductors, 

 we must consider such bodies as belonging to the shell in which 

 A is contained, and their surfaces as forming part of the surface 

 of B : in such cases this surface will generally be the interior 

 surface of the walls of the room in which A is contained, and 

 of all uninsulated conductors in the room. If, however, we 



* On the Uniform Motion of Heat, and its Connexion with the Mathe- 

 matical Theory of Electricity, vol. iii. p. 73 [Phil. Mag. S. 4. vol. vii. p. 502]. 



t The term used by Green for the sum of the quotients obtained by 

 dividing the product of each element of the surfaces of A and B, and its 

 electrical intensity, by its distance from P. 



E2 



