ACTIONS OF SPHERICAL LAYERS. 33 



hence the angles OPA and GAP' are equal ; calling r the 

 distance P'A, and D the distance OP, we have 



P = V 

 r R* 



Lastly, let du be the angle which subtends the element dS, seen 

 from the point P', 



dS cos a. 



Replacing dS cos a by this value in the expression for the com- 

 ponent <, we get 



R 2 

 y = " ^D 2 ' 



and the total action of the sphere is 



IT 47rR 2 o- M 

 "D 2 D 2 ' 



It is thus apparent that the action is the same as if the whole mass 

 M were concentrated at the centre of the sphere. 



The potential of the spherical layer on the outside is also the 

 same as if the whole mass were concentrated at the centre. 



If the point P is very near the surface, D = R and the action of 

 the layer is equal to 4770-, in agreement with what we have already 

 found (36) for any given conductor. 



The component, parallel to OP, of the action exerted by the 

 surface element dS, only depends on the angle da under which this 

 element is seen from the point P'. This component is then the 

 same for an element dS' situate at A', and opposed to the former in 

 reference to the point P'. 



The same is the case for all the elements of the zone CB'C' 

 compared in pairs with the elements of the layer CBC'. 



The plane CC' thus divides the surface of the sphere into two 

 parts, whose actions on the point P are equal, each of them being 



R 2 



equal to 2-rra- 



If the point P moves to an infinite distance, the two zones tend to 

 become equal ; if it is infinitely near the surface, the anterior zone 

 becomes infinitely small, and its action upon an infinitely near point 

 is reduced to 2770-. We have already obtained this result (40) for 

 any given surface. 



D 



