DISTRIBUTION OF ELECTRICITY ON CONDUCTORS. 



If the surface in question is a sphere, the angles / and /' are 

 equal. If, moreover, the sphere' is insulated, and not exposed to 

 any extraneous action, the distribution is homogeneous, and the 

 densities o- and </ are equal. 



The force at the point P is zero if the actions of the opposite 

 elements dS and dS' are equal, and for this it is sufficient if we have 

 the ratio 



= const, 



that is to say, that the electrical forces are inversely as the square of 

 the distance. 



The law of the square satisfies this condition, and is the only 

 one which does. This may be shown in a simple manner by the 

 following reasoning, which is due to M. Bertrand. 



Whatever /(r) may be, we may choose two values, r^ and r^ 



Fig. ii. 



such, that between these two values of the variable the product r 2 f(r) 

 always either increases or decreases as r increases. 



Let us construct a sphere (Fig. n) whose diameter is equal to the 

 sum r l + r zt and let us consider the point P which divides the 

 diameter into the two segments r^ and r y The actions of the opposite 

 elements dS and dS', determined by the same cone of aperture dw on 

 this point, are : for */S, 



cos t 



and for 



COS I 



All the values of r and r' are comprised between the limits r^ and 



