Electricity in Equilibrium. 4i7 



produced by the presence of such bodies in disturbing the 

 observed law of force, must be to make it diminish less rapidly 

 with the distance when A and B are separated by a considerable 

 interval : and it is probably owing, at least in part, to such dis- 

 turbing causes that Mr. Harris's results nearly agree, as far as 

 they go, with a formula which would ultimately give for the law 

 of force the inverse square of the distance between A and B, 

 instead of the inverse cube. 



4. The determination by the mathematical theory of the at- 

 traction or repulsion between two electrified conducting spheres 

 has not hitherto, so far as I am aware, been attempted, and 

 would present considerable difficulty by means of the formulse 

 ordinarily given for such problems. It may, however, very 

 readily be effected by means of a general theorem on the attrac- 

 tion between electrified conductors, which will be given in a 

 subsequent paper*. Thus, if F (c) be the force of attraction, 

 corresponding to the distance c between the centres, in the par- 

 ticular case when the two spheres are equal (the radius of each 

 being unity), and the potential in the interior of one of them 

 is nothing (as will be the case when the body is uninsulated), 

 the potential in the interior of the other being v, I have found 

 the following formulae which express F (c) by a converging 

 series. 



where 



^w=''^<lf^+§+S^+^4 • • (^) 



Ol =(c^-2)Q,-l, I ..... (B) 

 Q„+2=(c2-2)Qn+,-Q„. J 



P. =1, . 



Pg =2c2-3, V ..(C) 



P.4-2 = (c^-2)P,+j + (Q„+,-PJ.J . 



These formulse enable us to calculate Q^, Qg, Qg, Q^, &c., and 

 then P,, Pg, P3, P4, &c., successively, by a simple and uniform 

 arithmetical process, for any particular value of c. I have thus 



calculated the values of —V in five cases, the first four of 



which are those examined by Mr. Harris, and have obtained 



[* The enunciation of the " general theorem" aUuded to, the investiga- 

 tion founded on it, by which the author first arrived at the conclusion made 

 use of here, and another demonstration of the same conclusion, founded on 

 the method of electrical images, and strictly synthetical in its character, are 

 published, with comprehensive numerical results, in the Philosophical Ma- 

 gazine for April 1853.] 



