54 APPLICATION OF THE PRECEDING RESULTS 



and 2 



2 



Making r =a in this, and in the value of B' before given, in 

 order to obtain those which belong to the surface, there results 



f dB' B' _- 2a 2 + 2ab . cosfl + f _ V-a* 

 * dr' + a = af af 



This substituted in the general equation written above, there 



arses 



If P is supposed to approach infinitely near to the surface, 

 so that b = a a ; a being an infinitely small quantity, this 

 would become 



cc 



In the same way, by the aid of the equation between A and 

 p, the density of the electric fluid, induced on the surface of a 

 sphere whose radius is , when the electrified point P is exterior 

 to it, is found to be 



supposing the sphere to communicate, by means of an infinitely 

 fine wire, with the earth, at so great a distance, that we might 

 neglect the influence of the electricity induced upon it by the 

 action of P. If the distance of P from the surface be equal 

 to an infinitely small quantity a, we shall have in this case, as 

 in the foregoing, 



a 



o = - 



P 27T. 



From what has preceded, we may readily deduce the general 

 value of F, belonging to any point P, within the sphere, when 

 V its value at the surface is known. For (p), the density in- 

 duced upon an element do- of the surface, by a unit of electricity 

 concentrated in P, has just been shown to be 



y-a 2 



47m/ 3 ; 



