RECIPROCAL ACTION OF TWO SPHERES. 163 



these layers -is in equilibrium, their superposition will be a state of 

 equilibrium, and the resultant action will be equal to the resultant 

 action of all the images. The totality of these images will form a 

 system, which is the image, in reference to the sphere of the given 

 system. If the given system is a surface 2, the image will be a 

 surface conjugate to the first. 



176. RECIPROCAL ACTION OF Two SPHERES. The principle of 

 images combined with Murphy's method (86), enables us to solve 

 completely the very important problem of the reciprocal action of two 

 spheres. Let S a and S & be the two spheres (Fig. 48), R and R' the 

 radii. The method consists, as we know, in determining a series of 

 successive layers in the following manner. On the conductor S a a 

 layer is placed capable of giving the potential i ; this is a uniform 

 layer of mass R. This layer acts outwards as if it were concentrated 

 at A. It is fixed and the induced layer on the surface S & of the 



Fig. 48. 



second uninsulated sphere is determined, which amounts to deter- 

 mining the image A' in reference to S 6 of a mass + R at A. The 

 equivalent layer is next fixed at A', and its inductive action on the 

 uninsulated sphere S a is determined that is to say, the new image A l 

 of A', and so on. The same operation will be repeated beginning 

 with the sphere S 6 , and all the masses thus determined are multiplied 

 by suitable coefficients. As each of the masses and the densities 

 can be exactly calculated, the problem of distribution is completely 

 solved. 



The force exerted between the two spheres is the resultant of the 

 actions exerted by each of the masses comprised within one of the 

 spheres on all the masses contained in the second. 



M 2 



