124 



It is to be observed that k v A- 2 , &c., tar/'), w,, w^\ &c. will be 

 functions of x if the section of the conducting system is hetero- 

 geneous in different positions along it ; but in all cases in which each 

 conductor is uniform, and uniformly situated with reference to the 

 others along the whole length, these coefficients will be constant, and 

 the equations become reduced to 



. . (5). 



dt ~ k, dx* k, da* r A 3 dJ + &C ' 



dq^_ 



dt ~ 



The most obvious general method of treatment for integrating 

 these equations, is to find elementary solutions by assuming 



q t =A t u, q. 2 =A 2 u, <? 3 = A 3 w, ..... ?i=A i w, . . (6), 

 where u satisfies the equation 



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

 tions among the coefficients A 1} A 2 , . . . . A f , giving by elimination 

 an algebraic equation of the ith 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 



