358 On the Attraction of Ellipsoids, 



Now, in order to obtain the attraction of the whole ellipsoid 

 on the point C, we have to integrate the expression given above 

 between the limits u = and u = 1, or OK = OA,, and OK = OKj ; 

 from which it appears that its value is 



47r,u/p abc 1 



(rt 2 - C 2 ) (6 2 - C 2 ) 



It is easy to see that the attraction of the part of the ellipsoid 

 contained within the conical surface, whose semi-angle 9 is equal 

 to the angle cos" 1 u, is 



where t = tan KQ - arc A U. 



To represent the attraction on a point n placed at the extre- 



mity of the mean axis, assume on the transverse OA of the focal 



g 

 hyperbola a point KI such that OK! = OA r, and from KI 



draw a tangent KjQi to the hyperbola, and let T = tan K^ 

 - arc AoQi ; then the attraction of the ellipsoid on the point /j. is 



70 



T ' 4 



To prove this, assume a point K such that 



OA 



OK = ~V(^ + 2 (c 2 -r)}; 



from K draw a tangent KQ, to the hyperbola, and from let 

 fall a perpendicular OP on this tangent; then if i// = angle 



A () OP, 



(a 2 - b 2 } u* 



sin 2 !// = 



(a 2 - 



Hence, by following a method similar to that used in finding the 

 representation of the attraction on a point at the extremity of 

 the least axis, the expression given above may be easily obtained. 

 The attractions (7, B of the ellipsoid on points placed at the 

 extremity of the least and mean axes are thus represented by 



