296 Prof. Thomson on Electrified Spherical Conductors. 



found by using the form (/) or (18). But if we add the terms 

 in the horizontal lines, we find the following convergent series 

 forF:— 



[ ^,iog^.^^^ ^iiogl.^v^ ^Ml.&^de \ 



Hence, since (H-(9)-2=l-2^-f-3^-&c., we have 



^=^:/:^ 



,log^ 

 or, by actual integration, 



=i;«.g X (^log2- j)=z;^i x (•69315--25) 



= i;2x -073858. 



The quantity of electricity on each sphere being equal to the 

 sum of the masses of the imaginary series of points within it, is, 

 according to the formulae for pi, q\y p^, q\y kc. 



v(^l-^ + g-j + &c.j, or v log 2. 



Hence we have the following expression for the repulsion between 

 the two spheres, in terms of Q the quantity of electricity on each, 



*-^- (log 2)^ • 



If x denote the distance at which two electrical points, con- 

 taining quantities equal to the quantities on the two spheres, 

 must be placed so as to repel one another with a force equal to 

 the actual force of repulsion between the spheres, we have 



(t;.log2)^ _ 

 Using the value for F found above, we obtain 



log 2 O C-A 



^{^x(log2-i)} 



If the electrical distributions on each surface were uniform, this 

 distance would be equal to 2, the distance between the centres 

 of the spheres ; but it exceeds this amount, to the extent shown 

 by the preceding result, because in reality the electrical density 



