between two Electrified Spherical Conductors. 393 



Also we have directly from (e) and (c). 



'~2ab' ^~2 a'b^ 

 ■pi _n pi __£_ 



f 



(h). 



These equations enable us to calculate successively the values 

 of S'l, S'^, S'g, &c., P\, V'^ P'e, &c., and Q'„ Q'^, Q'3, &c., 

 after the values of Sj, Sg, &c.^ Pp Pg, &c.j and Qi, Q2, &c. have 

 been found. 



The solution of [b) as equations of finite differences with 

 reference to n, and the determination of the arbitrary constants 

 of integration by [c], leads to general expressions for S^, P^^ 

 and Q^; and by using these in {g), integrating the equations 

 so obtained, and determining the arbitrary constants by means 

 of [Ji)j general expressions for S'„, P'^, and Q'„ are obtained. 

 The expression for F may therefore be put into the form of an 

 infinite series, with a finite expression for the general term. 

 Further, the value of this series may be expressed, by means of 

 analysis similar to that which Poisson has used for similar pur- 

 poses, in terms of a definite integral. I do not, however, in 

 the present communication give any of this analysis, except for the 

 case of two spheres in contact which is discussed below, because, 

 except for cases in which the spheres are very near one another, 

 the series for F is rapidly convergent, and the terms of it may 

 be successively calculated with great ease, by regular arithmetical 

 processes, for any set of values of c, a, and h, by using first the 

 equations (c), to calculate Sj, Sg, Pp Pg, Q^, Qg; then {b) 

 with the values 2, 3, &c. successively substituted for n, to calcu- 

 late S3, S4, &c., and P3, P4, &c., and Q3, Q4, &c. ; then (A) and 

 {g) to calculate by a similar succession of processes, the values 

 of S'l, S'2, S'g, &c., P'l, P'„ P'3, &c., and Q'„ Q'„ Q'3, &c. 



The following is the method, alluded to above, by which I 

 first arrived at the solution of this problem in the year 1845. 



The "mechanical value ^■' of a distribution of electricity on a 

 group of insulated conductors, may be easily shown to be equal to 

 half the sum of the products obtained by multiplying the quan- 

 tity of electricity on each conductor into the potential within it*. 



* This proposition occurred to me in thinking over the demonstration 

 which Gauss gave of the theorem that a given quantity of matter may be 

 distributed in one and only one way over a given surface so as to produce a 

 given potential at every point of the surface, and considering the mechanical 

 signification of the function on the rendering of which a minimum that de- 



