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



Let E' be another ellipsoid contiguous to E l and inside it and confocal 

 with both E and E^; let its principal semi-axes be of, b', c', and let 



then since c^ 2 a'- = b* b'- = c? c' 2 , 



we have ultimately a'Sa' = b'Sb' c'Sc'. 



Imagine a shell of which E' is the inner boundary and containing 

 the same quantity of matter as E or E u and let this matter be of uniform 

 density, then the potential of the shell E at any point on the surface of 

 E' is constant and equal to the potential of shell E' at the common 

 centre. 



Now let S be the sphere whose centre is at O and radius unity. 

 Imagine a shell of which the inner boundary is S ; let I, m, n be the co- 

 ordinates of any point p on S, and let So- be an element of the surface at 

 p ; then if a cone be described with base Scr and vertex O, the element 

 of the shell S intercepted : whole volume of shell : : Scr : 4?r. 



At the points P 1 on E l and P' on E', which correspond, take elements 

 of the respective shells which correspond to the element at p on this spherical 

 shell. 



The volumes of these corresponding elements will be proportional to the 

 whole volumes of the shells to which they belong, hence if M denote the 

 mass of each of the shells E, E l and E', the mass of the element at P l 



M 



and also at P' will be . Scr ; also the coordinates of P l are aJ, b,m, cyi 



4?r 



and those of P' are a'l, b'm, c'n ; 



therefore OPf - OP H = I 2 (a? - a' 2 ) + m 3 (b? - b" 2 ) + iv (c, 2 - c' 2 ) 



= (a, 2 - a' 2 ) (I- + m + n 2 ) = a, 2 - a' 2 . 



Let OP 1 = r 1 and OP' = r' and let r^r' + Sr'; then we have 



r'Sr' = a'Sa'. 



Now if V be the potential of the shell E l at 0, and V = V+ S V be 

 the potential of the shell E' at the same point, then 



M (do- 



V= 



47T I 1 



cr , TT> M fdcr 

 - and V'=- -j-; 

 \ 47T J r 



