' Electricity in Equilibrium. ' *^>*'^ 63 



The problem in each case is determinate, and we may therefore 

 employ the elementary principles of one theory, as theorems, 

 relative to the other. Thus, in the paper in which these consi- 

 derations are developed, CoulomVs fundamental theorem relative 

 to electricity is applied to the theory of heat ; and self-evident 

 propositions in the latter theory are made the foundation of 

 Green's theorems in electricity*. Now the laws of motion for 

 heat which Fourier lays down in his Theorie Analytique de la 

 Chaleurj are of that simple elementary kind which constitute 

 a mathematical theory properly so called ; and therefore, when 

 we find corresponding laws to be true for the phsenomena pre- 

 sented by electrified bodies, we may make them the foundation 

 of the mathematical theory of electricity : and this may be done 

 if we consider them merely as actual truths, without adopting 

 any physical hypothesis, although the idea they naturally sug- 

 gest is that of the propagation of some efi^ect by means of the 

 mutual action of contiguous particles; just as Coulomb, although 

 his laws naturally suggest the idea of material particles attract- 

 ing or repelling one another at a distance, most carefully avoids 

 making this a physical hypothesis, and confines himself to the 

 consideration of the mechanical efi'ects which he observes and 

 their necessary consequences f. 



All the views which Faraday has brought forward, and illus- 

 trated or demonstrated by experiment, lead to this method of 

 establishing the mathematical theory, and, as far as the analysis 

 is concerned, it would, in most general propositions, be even 

 more simple, if possible, than that of Coulomb. (Of course the 

 analysis of particular problems would be identical in the two 

 methods.) It is thus that Faraday arrives at a knowledge of 

 some of the most important of the general theorems, which, 

 from their nature, seemed destined never to be perceived except 

 as mathematical truths. Thus, in his theory, the following 

 proposition is an elementary principle. Let any portion a of 

 the surface of A be projected on B, by means of lines (which 

 will be in general curved) possessing the property that the 

 resultant electrical force at any point of each of them is in the 

 direction of the tangent : the quantity of electricity produced 

 by induction on this projection is equal to the quantity of the 

 opposite kind of electricity on a J. The lines thus defined are 

 what Faraday calls the " curved lines of inductive action.^' For 



■ * It was not until some time after that paper was published, that I was 

 able to add the dh-ect analytical demonstrations of the theorems, which 

 are given in the papers on " General Propositions in the Theory of Attrac- 

 tion," Math. Journ. vol. iii. pp. 189, 201, and which I have since found, 

 are the same as those originally given by Green, 

 t See Note I. + See Note IV. ^ 



