between two Electrified Spherical Conductors. 



295 



The series (/) or (18) for F becomes divergent for the case 

 of two spheres in contact, but the doubly infinite series from 

 which this was derived in the first of the two investigations 

 given above, is convergent when the terms are properly grouped 

 together ; and its sum may be expressed by means of a definite 

 integral in the following manner. 



Since the two spheres are in contact, the potentials within 

 them must be equal, that is, we must have u=v. For the sake 

 of simplicity, let us suppose the radii of the two spheres to be 

 equal, and let each be taken as unity. Then we shall have 

 a = b = \, and c=3; and the terms of doubly infinite series (9) 

 in this case are easily expressed*, in very simple forms, by equa- 

 tions (8). Thus we find 



F = i;2x 



22 



1 



32 



2.1 



2 1.3 



1 



32 



+ 



42 

 2.2 



52 

 2.3! 



Si 



42 

 3.1 



52 

 3.2 



4- 



+ 



52 

 4.1 



52 



+ 



62 

 2.4 



6^ 

 3.3 



62 

 4.2 



62 

 5.1 



62 



&c. 



-&c. 



~&c. 



-&c. 



-&c. 



w 



If we add the terms in the vertical columns, we find 

 2.3 2.3.4 3.4.5 



'— K^ 



4 3.4 



&c.), 



22 32 ' 42 



which is a diverging series, and is the same as we should have 



From equations (8) we find, in this case. 



Hence 



fn — g n — 



2n-\ 



2n 



V 



'2^' 



fn — C/n — 

 pn=qn 



p'n=^q'n = 



Ps9t _ (2s-l)(2t-\) 

 (c-f,-fftf^{2{s+t)-2\^ 



Pj9't _ 2s. 2t 



ic-f^ 



PsP't 



2n—2 

 2n-l 



V 



2w~l 



■9't)' 



Qs'i't 2t(2s-l) 



(o-f-AY {c-9-9',) {2{s-^t)-.\}' 



•and then, hy (9), we obtain the expression for F in this particular case, 

 given in the text. 



