34 ON POTENTIAL. 



44. ACTION OF A SPHERE CONSISTING OF HOMOGENEOUS 

 LAYERS. Let us first consider a sphere electrified throughout its 

 whole mass, and made up of homogeneous concentric layers. 



The action of this sphere on an external point is the same as 

 if the whole mass were concentrated at the centre, or were carried 

 to the surface so as to form a uniform layer. 



On a point in the interior of the sphere the action of the layers 

 which surround it is null ; that of the layer whose radius is less 

 than its distance from the centre is still the same as if the mass 

 were concentrated at the centre. 



Proceeding towards the centre the acting mass diminishes then 

 more and more; the direction of the force is always along the 

 radius, and depends on the manner in which the density varies. 



When the density p of the sphere is constant, the action exerted 

 on a point in the interior, at the distance r t is equal to 



it is thus proportional to the distance from the centre. 



If the density at a point is proportional to the n th power of its 

 distance / from the centre, p = al n ; the action of the layer dl at the 

 distance r is 



and the total action 



n 



n + s 



This action is constant for n = - i ; it increases, on the contrary, 

 approaching the centre, if n < i. 

 The total mass of the sphere of radius R is 



47T# 



which gives 



r 



M= ^irfial 



J 



Rn+3 



Let us suppose that the density varies according to an arbitrary 

 law. Let m be the mass external to the sphere which passes through 



