53] ON THE ATTRACTION OF AN INDEFINITELY THIN SHELL. 421 



therefore the resolved part of the attraction of the element E at P along 

 M t M is pj - , . Sco', therefore the attraction of the whole shell on M l along 



Otjt/jCj 



M,M will be 



Hence if the shell be of uniform density />, the attraction of the 

 whole shell on M l in the direction of the normal will be Trptp, . 



C^OjC, 



where p 1 is the perpendicular from the origin on the tangent plane at M v 



Hence the attraction of the shell has been determined in direction and 

 magnitude. 



Second Solution. 



Imagine a shell of which E is the inner boundary to be composed 

 of matter of uniform density, and another shell of which E l is the inner 

 boundary to contain the same quantity of matter, also of uniform density. 

 The quantity of matter contained in any portion of E will be equal to 

 that in the corresponding part of E,, 



also since vol. of E : vol. of E l :: abc : a^^; 



therefore density of E : density of E l :: a 1 b l c 1 : abc. 



Now let M' and M 1 be two fixed corresponding points on E and E l , 

 and let P and P l be any two corresponding points ; then by the property 

 of corresponding points on confocal ellipsoids, M'P r = Mf. 



Also the same quantity of matter is contained in corresponding elements 

 of the two shells at P and P lt and since the same is true for all cor- 

 responding elements, therefore the potential of shell E l at the point M' 



= the potential of shell E at the point M^. 



But since, by Newton's Theorem, the shell E 1 exerts no attraction on 

 an internal point, its potential is constant at all internal points and is 

 therefore the same at M' as at O, the common centre of E and JH 1 . 



Hence the potential of the shell E at any point M l on the surface 

 of E l is constant and equal to the potential of the shell E 1 at its centre 

 ; therefore by the theory of the Potential the attraction of the shell E 

 at M 1 is in the direction of the normal to the surface 2?,. 



We now proceed to find the magnitude of this attraction. 



