222, 223] Images 195 



from which, in the same way, 



The value of q<& can of course be written down by symmetry from that of q n . 

 The coefficients each depend on a sum of the type 



This series cannot be summed algebraically, but has been expressed as a definite integral 

 by Poisson. From the known formula 



we obtain at once 



r_sjn* fe*M-n 1 



J e -i * V-1J ~2p' 



1 _i 1 2 f sin^ 



r^~ '~7 Jo e=i att 



so that on putting p = log 2 a 28 we have 



8 _ ! . a 9 

 l- 2 a 2 ~ log ?a*~ Jo e 2 ^-! 



From this follows 



8 r S a 8 sin (2 log g+2s log a) t . 



Both the series on the right can be summed. We have 



a = 

 so that 2 _ = _L_ 





_ 



Jo (e - 1) (1 - 2a cos (2# log a) + a 2 ) 

 and on replacing by unity, we obtain 



^ 8 1 r asin(2nogq) ' 



1-a 28 2(l-a)" 1 " J (e 27r< -l)(l-2aCOS(2Hoga)+a 2 ) 



These are the series which occur in q n and q^. 



223. Having calculated the coefficients, either by this or some other 

 method, we can at once obtain the relations between the charges and poten- 

 tials, and can find also the mechanical force between the spheres. If this 

 force is a force of repulsion F, we have 



oragan . 



132 



