114 116] ATTRACTION OF ELLIPSOIDS. 217 



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



c 2 a 2 



This is Ivory's theorem : Two con focal 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.* 



116. Ellipsoids of Revolution. For an ellipsoid of revolu- 

 tion, the elliptic integrals reduce to inverse circular functions. 



Put b = c, a being the axis of revolution, 



r du 



(i) F=7rao 2 . UXx + Yy), 



J<r(b* + u)\/a 2 + u 



where - + , ^ = 1. 



a 2 + a- b z + a 



Put 



} 

 s 



, 

 du = --- ^ - -- - ds. 



o 



/^ 2 _ g2 



When u = oo , s = ; when u cr, s = A/ ^ - , so that 



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



