420 ON THE ATTRACTION OF AN INDEFINITELY THIN SHELL. [53 



Hence if p = l, the attraction of the element on M, resolved in the 

 direction 



.. cosPM,M_pt.Su.MP s M,P cos , 



MlM ~ MP cos MPN ' ~ M.P* M.P 3 ' MP cos MPN ' 



Let x, y, z be the coordinates of P, then the direction cosines of PN 



are ^ , ^r , *-= and the proiection of MP upon the normal PN will be 

 a- b~ c- 



px I 2 \ p y I b- \ pz 

 - -- 



Z :' 



a' \ <v 7 i" 



or MP cos MPN=-p I 1 - fe + p" + fOT 



Similarly MjP cos PM,M is the projection of M t P upon 



The direction cosines of M.M are -- 1 -,- 1 , *TT. l . , where , is the per- 



,- 6,- c,- 



pendicular from origin on the tangent plane at M r 

 The projection of Mf upon M^M is 



nri np 01 -77 'ii v i f V tl Y t mil 



//I lA/j . v JL/I f/i / v t \ \ / \ / **" j t'i */*/i 



C^i t/-, Oi \ (6, O-i 



Hence attraction of element at P on Jf, resolved in the direction M^M is 



MP 3 

 t.8o>. p, -jg-p = t.8u.p l ^-p ;i (since M,P = M'P'). 



Let Sw' be the solid angle of a cone whose vertex is M' and base 

 the element of E' which corresponds to the element E at P. 



Then the volume of this cone is ultimately - Sw' . M'P' 3 . 



But the volume of the corresponding cone will be -Sat. MP 3 , and 



O 



these volumes are as aj^ : abc respectively ; 

 therefore 



. , 



M/P/3 =00) . 



abc 

 abc 



