52 APPLICATION OF THE PRECEDING RESULTS 



value of the density p will be immediately given, by one or other 

 of these equations. 



From what has preceded, we may readily determine how the 

 electric fluid will distribute itself, in a conducting sphere whose 

 radius is a, when acted upon by any bodies situate without it ; 

 the electrical state of these bodies being given. In this case, 

 we have immediately the value of the potential function arising 

 from them. Let this value, for any point p within the sphere, 

 be represented by A ; A being a function of the radius r, and 

 two other polar co-ordinates. Then the whole of the electricity 

 will be carried to the surface (art. 1), and if Fbe the potential 

 function arising from this electrified surface, for the same point 

 p, we shall have, in virtue of the equilibrium within the sphere, 



V+A=ft or V=ft~A; 



ft being a constant quantity. This value of V being substituted 

 in the first of the equations (9), there results 



n dA A $ 

 47T/0 = - 2 -7 --- +-: 

 ar a a 



the horizontal lines indicating, as before, that the quantities 

 under them belong to the surface itself. 



In case the sphere communicates with the earth, ft is evi- 

 dently equal to zero, and p is completely determined by the 

 above : but if the sphere is insulated, and contains any quantity 

 Q of electricity, the value of ft may be ascertained as follows : 

 Let V be the value of the potential function without the surface, 

 corresponding to the value V = ft A within it; then, by what 

 precedes 



A' being determined from A by the following equations : 



and r', being the radius corresponding to the point p', exterior 



