ON POTENTIAL. 



?* 2 , and the value of r is always less than the corresponding value 

 of r' ; the value of r*f(r) for the elements of the upper portion will 

 always be smaller or larger than that of r' 2 f(r') for those of the lower 

 portion ; the action of the zone D AC is therefore smaller or larger 

 than that of the zone DEC. Equilibrium would thus be impossible 

 unless r*f(r) is a constant that is, unless the function f(r) were 

 exactly in the inverse ratio of the square of the distance. 



43. ACTIONS OF SPHERICAL LAYERS. The action of a homo- 

 geneous spherical layer on an external point is the same as if the whole 

 mass were concentrated at the centre of the sphere. 



For let us consider the action which a sphere S (Fig. 12), covered 

 with a homogeneous layer of density <r, exerts upon an external 

 point P. 



Fig. 12. 



The action being obviously directed towards the centre, it will be 

 sufficient if we take the sum of the components of the elementary 

 actions in this direction. 



This component for the surface element dS at the point A, which 

 is at a distance p from P, is 



<p = COS a. 

 P 2 



Let P' be the conjugate of the point P; that is to say, such that 

 OP'.OP = R 2 ; 



if we draw AP' and AO, the triangles AOP' and AOP are similar, 

 for the angle at O is common, and we have the ratio 



OF R 



