138 Royal Society : — 



. (5). 



The most obvious general method of treatment for integrating 

 these equations, is to find elementary solutions by assuming 



q^=■^^u, q.^ = A,u, q^ = K^u, qi=^i^ ■ • (^). 



where u satisfies the equation 



du d-u * /MX 



- — = X - (.7)' 



dt dx^ 



This will reduce the differential equations (5) to a set of linear equa- 

 tions among the coefficients Aj, A^, . . . . A;, giving by elimination 

 an algebraic equation of the 2th degree having i real roots, to deter- 

 mine X. The particular form of elementary solution of the equation 

 (7) to be used may be chosen from among those given by Fourier, 

 according to convenience, for satisfying the terminal conditions for 

 the different wires. 



In thinking on some applications of the preceding theory, I have 

 been led to consider the following general question regarding the 

 mutual influence of electrified conductors: — If, of a system of de- 

 tached insulated conductors, one only be electrified with a given ab- 

 solute charge of electricity, will the potential excited in any one of 

 the others be equal to that which the communication of an equal 

 absolute charge to this other would excite in the first 1 I now find 

 that a general theorem communicated by myself to the Cambridge 

 Mathematical Journal, and published in the Numbers for November 

 1842 and February 1843, but, as I afterwards (Jan. 1845) learned, 

 first given by Green in his Essay on the Mathematical Theory of 

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

 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 Np N.^ 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„ 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 //"NiVj^/s is the same as that of J^N^V ids 

 (each being equal to the integral yVy^iI^a^i" ^'^'^^i"^^ extended 

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

 (x, y, 2) of which external space the two resultants are denoted by 

 Rj, Ro respectively, and the angle between their directions by 6). 

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



