50 APPLICATION OF THE PRECEDING RESULTS 



the whole of that originally contained in the shell itself, and in 

 all the interior bodies. 



This is so direct a consequence of what has been shown in 

 articles 4 and 5, that a formal demonstration would be quite 

 superfluous, as it is easy to see, the only difference which could 

 exist, relative to the interior system, between the case where 

 there is an exterior system, and where there is not one, would 

 be in the addition of a constant quantity, to the total potential 

 function within the exterior surface, which constant quantity 

 must necessarily disappear in the differentials of this function, 

 and consequently, in the values of the attractions, repulsions, and 

 densities, which all depend on these differentials alone. In the 

 exterior system there is not even this difference, but the total 

 potential function exterior to the inner surface is precisely the 

 same, whether we suppose the interior system to exist or not. 



(10.) The consideration of the electrical phenomena, which 

 arise from spheres variously arranged, is rather interesting, on ac- 

 count of the ease with which all the results obtained from theory 

 may be put to the test of experiment ; but, the complete solution 

 of the simple case of two spheres only, previously electrified, and 

 put in presence of each other, requires the aid of a profound 

 analysis, and has been most ably treated by M. PoiSSON (Mem. 

 de 1'Institut. 1811). Our object, in the present article, is merely 

 to give one or two examples of determinations, relative to the 

 distribution of electricity on spheres, which may be expressed by 

 very simple formulae. 



Suppose a spherical surface whose radius is a, to be covered 

 with electric matter, and let its variable density be represented 

 by p ; then if, as in the Me*c. Celeste, we expand the potential 

 function V, belonging to a point p within the sphere, in the 

 form 



r being the distance between p and the centre of the sphere, and 

 U {0 \ U (1 \ etc. functions of the two other polar co-ordinates of p, 

 it is clear, by what has been shown in the admirable work just 



