GENERAL PRELIMINARY RESULTS. 41 



Hence the equation determining p becomes, by substituting for 



its value just given, 



[p^_^fd^(idV\ (dV\\ 

 ] r "47rJ r |U*/ W/J' 



an equation which could not subsist generally, unless 



dV 



Thus the whole difficulty is reduced to finding the value V t of 

 the potential function exterior to the body. 



Although we have considered only one body, it is clear that 

 the same theory is applicable to any number of bodies, and that 

 the values of p and p will be given by precisely the same for- 

 mulae, however great that number may be: V t being the ex- 

 terior potential function common-to all the bodies. 



In case the bodies under consideration are all perfect con- 

 ductors, we have ^eeii (art. 1), that the whole of the electricity 

 will be carried to their surfaces, and therefore there is here no 

 place for the application of the theory contained in this article ; 

 but as there are probably no perfectly conducting bodies in 

 nature, this theory becomes indispensably necessary, if we would 

 investigate the electrical phenomena in all their generality. 



Having in this, and the preceding articles, laid down the 

 most general principles of the electrical theory, we shall in what 

 follows apply these principles to more special cases; and the 

 necessity of confining this Essay within a moderate extent, will 

 compel us to limit ourselves to a brief examination of the more 

 interesting phenomena. 



