164 PARTICULAR CASES OF EQUILIBRIUM. 



The calculation does not present any theoretical difficulties, but 

 it is very tedious. Sir W. Thomson performed it in the case of two 

 spheres of the same radius when the distance of the centres varies 

 between 2R and 4R, that is to say, when the distance of the surfaces 

 is comprised between and the diameter of one of the spheres. 



In the present case, if R is the common radius of the spheres 

 A and B, U and V the potentials, <rR the distance of the centres, M 

 and N the respective charges, then if I, J, a and b are coefficients 

 which depend on c, we have 



M = R(IU-JV), 

 N = R(IV-JU), 



expressions analogous to those furnished by Murphy's method for any 

 given bodies. 



If we wish to express the force, and the potential as a function 

 of the masses, we get 



E. 2 F = 2/3MN - a(M 2 + N 2 ), 



_ 



(I2 _J 2)2 



( 



(I 2 -J 2 ) 2 



If the charges M and N are equal, we get 

 M = RV(I-J), 



R 2 F = 2 (/3-a)M 2 . 



177. As these formulae have only hitherto been calculated for 

 c = 4, it is useful to see how they may be replaced for greater distances. 



