125 



Electricity and Magnetism (Nottingham, 1828), leads to an affirmative 

 answer to this question. 



The general theorem to which I refer is, that if, considering the 

 forces due respectively to two different distributions of matter (whether 

 real, or such as is imagined in theories of electricity and magnetism), 

 we denote by N^ N 2 their normal components at any point of a closed 

 surface, or group of closed surfaces, S, containing all parts of each dis- 

 tribution of matter, and by V,, V 2 the potentials at the same point 

 due respectively to the two distributions, and if ds be an element of 

 the surface S, the value of J^^^^ds is the same as that ofJ^fN 2 V l ds 

 (each being equal to the integral /VY^R^ sin 0afa?dycfe extended 

 over the whole of space external to the surface S, at any point 

 (x, y, z) of which external space the two resultants are denoted by 

 Bj, R 2 respectively, and the angle between their directions by 0). 

 To apply this with reference to the proposed question, let the first 

 distribution of matter consist of a certain charge, q, communicated to 

 one of a group of insulated conductors, and the inductive electrifica- 

 tions of the others, not one of which has any absolute charge ; let 

 the second distribution of matter consist of the electrifications of the 

 same group of conductors when an equal quantity q is given to a 

 second of them, and all the others are destitute of absolute charges ; 

 and let surface She the group of the surfaces of the different conductors. 

 Since the potential is constant through each separate conductor, the 

 integral /yN^j ds will be equal to the sum of a set of terms of the 

 form [V,][^jfN ,<?], where [V 2 ] denotes the value in any of these 

 conductors of the potential of the second distribution, and [/yN,^*] 

 an integral including the whole surface of the same conductor, but 

 no part of that of any of the others. Now by a well-known theorem, 

 first given by Green, [ /"/"Njcfe] is equal to 4ifq if q denote the abso- 

 lute quantity of matter within the surface of the integral (as is the case 

 for the first group of conductors), and vanishes if there be no distri- 

 bution of matter, or (as is the case with each of the other conductors) 

 if there be equal quantities of positive and negative matter within the 

 surface over which the integral is extended. Hence if [V^! denote 

 the potential in the first conductor due to the second distribution of 

 matter, we have 



