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



We may regard a homogeneous ellipsoid as made up of indefinitely 

 thin shells. 



Let X, Y, Z be the components in the direction of the axes of the 

 attraction of an ellipsoid whose semi-axes are a, b, c on the point P lt 

 and let X + BX, Y+BY, Z+BZ be the attractions of a similar ellipsoid whose 

 semi-axes are a + So, b + Bb, c + Be, where 



Ba at, Bb bt, Be = ct, 



then 



l-rrpabc 

 



bc }\x kirpbc p* 



. tp, . 7 -V = , r . . x . Ba. 



- , . 

 of o,c, a, 



Let u , then Bu = . Ba, ultimately, 

 and a l B(f l = 2\'t, 



hence Bu = - 3 .p?. Ba; 



hence 8A' = 4^-n-px . , . Bu, 



2^ be of ? 



OJ. = +TTp!/ , TT "> 



s,~ be of 1 



O/j = 4:TTpZ . , . 7, . OU. 



be a ~ 



We have now to substitute for the quantities 7 , &c. 



6,0, 6f 



Since a* a~ = b{ b~ = c? c~, 



the equation to the ellipsoidal shell through the attracted point is 



X' y" 



j ^ i 



. 1 T . ., , /7 a .ox I 



a? a* + (b~ a 2 ) a, 2 + (c 2 a 2 ) 



and so we get 



of if 



I .y i 



where - and are constants ; and so a 2 is known in terms of u\ 



a a 



Also &,' = a" [~- 2 + (-.- l\] and c, 2 = a 2 [ -. + (*. - l}~\ . 



LU \a- /J [u- \a- ]] 



