between two Electrified Spherical Conductors. 289 



the centre of B. To specify completely these two series of ima- 

 ginary points_, let p^, p\, p^, p\, p^, pK^, &c. denote the masses of 

 the series of which the first is at the centre of A ; and let f^, /j, 

 /s) f<2} &c. denote the distances of these points from the centres 

 of A and B alternately ; and, again, let q^, q\, g-g, §'2 • • • denote 

 the masses, and g^, g\, g^^ 9\ - - > the distances of the successive 

 points of the other series from the centres of B and A alter- 

 nately. These quantities are determined by using the following 

 equations, and giving n successively the values 1, 2, 3, ... ; 



^ J n 



a 



4=00 __,C- = 





+ 



<i.<it 



f^{p-n-9iY^{c-f-f<) 



+ 



{•^-g-^y^ 



(8). 



The two series of imaginary electrical points thus specified^ 

 would, if they existed, produce the same action in all space 

 external to the spherical surfaces as the actual distributions of 

 electricity do, those [p^, g'vP^} 9'<2) ^^•) which lie within the 

 surface A producing the effect of the distribution on A, and 

 the others {q^, p\, q^, p'^, &c.), all within the surface B, the 

 effect of the actual distribution on B. Hence the resultant 

 force between the two partial groups is the same as the resultant 

 force due to the mutual action between the actual distributions 

 of electricity on the two conductors; and if this force, considered 

 as positive or negative according as repulsion or attraction pre- 

 ponderates, be denoted by F, we have 



[ • • (9), 



where SS denotes a double summation, with reference to all 

 integral values of s and t. The following process reduces this 

 double series to the form of a single infinite series, of which the 

 successive terms may be successively calculated numerically in 

 any particular case with great ease. 



First, taking from (8) expressions for ps and fg in terms of infe- 

 rior order, and for qt and gt in terms of higher order, and con- 

 tinuing the reduction successively, we have 



