TO THE THEORY OF ELECTRICITY. 57 



We therefore see, that when the distance Z> between the centres 

 of the spheres is very great, the mean densities will be inversely 

 as the radii ; and these last remaining unchanged, the density 

 on the smaller sphere will decrease, and that on the larger 

 increase in a very simple way, by making them approach each 

 other. 



Lastly, let us endeavour to determine the law of the distri- 

 bution of the electric fluid, when in equilibrium on a very thin 

 spherical shell, in which there is a small circular orifice. Then, 

 if we neglect quantities of the order of the thickness of the shell, 

 compared with its radius, we may consider it as an infinitely 

 thin spherical surface, of which the greater segment S is a 

 perfect conductor, and the smaller one s constitutes the circular 

 orifice. In virtue of the equilibrium, the value of the potential 

 function, on the conducting segment, will be equal to a constant 

 quantity, as F, and if there were no orifice, the corresponding 

 value of the density would be 



a being the radius of the spherical surface. Moreover on this 

 supposition, the value of the potential function for any point P, 



W 



within the surface, would be F. Let therefore, - + p re- 



4?ra 



present the general value of the density, at any point on the 

 surface of either segment of the sphere, and F+ V, that of the cor- 

 responding potential function for the point P. The value of the 

 potential function for any point on the surface of the sphere will 

 be F+ V, which equated to F, its value on $, gives for the 

 whole of this segment 



0=F. 

 Thus the equation (10) of this article becomes 



the integral extending over the surface of the smaller segment 

 s only, which, without sensible error, may be considered as a 

 plane. 



