158, 159, 160] IVORY'S THEOREM. 421 



These will be called corresponding points. We shall now assume 

 that these two ellipsoids are confocal, and (2) the smaller. Then 



49) V = a 2 2 + A, V = V + *, V = <*' + * 



The action of (2) on the external point x, y, z is 



50) X 2 = - 2<yxa 2 ~b 2 c 2 x C du 



J ( 



a 



where 



x * I V* | 

 a^+6^ V+o 

 and since 



* , *!' , > 



v + v + 



we must have (5 = 'k. 



If now we substitute 



51) X 2 = 2y7taJ)<,Cc>x I - u 



J (i 2 + ') V(oi 2 + ') 

 o 



Now the attraction of the ellipsoid (1) on the interior point 

 5 c 2 . 



> y^> 2 is 



a x f-.^ q 



cc 



52) Zj = - 2 7 7ca 1 b 1 c L x^ C - du 



<V KH^l/^^ + ^^H^^H^) 







The definite integrals being the same in both cases, we have 



^ = M i 

 X, \c,' 



53) ^ 



This is Ivory's theorem: Two confocal ellipsoids of equal density 

 each act on corresponding points on the other with forces whose 

 components are proportional to the areas of their principal sections 

 normal to the components. 1 ) 



16O. Ellipsoids of Revolution. For an ellipsoid of revolution, 

 the elliptic integrals reduce to inverse circular functions. 



1) Ivory, " On the attractions of homogeneous Ellipsoids." Phil. Trans., 1809. 



