Professor Mac Cullagh's Lectures on the Attraction of Ellipsoids. 385 



wliere t = tan KQ — arc Ao Q. 



To represent the attraction on a point /t placed at the extremity of the 

 mean axis, assume on the transverse axis OAo of the focal hyperbola a point 



(^ 

 Ki such that OKj = OAo ,, and from K, draw a tangent Kj Qi to the hyper- 

 bola, and let T = tan Ki Q, — arc Ao Qi, then the attraction of the ellipsoid on 

 the point n is 



OA 



To prove this, assume a point K such that OK = —j-^ \/\b' + u- (c^ — ^■)j ; 



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

 cular OP on this tangent, then if i^ = angle AoOP, 



sm- -v^ = 



b" + ti-{a^- b-y 



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 C, B of the ellipsoid on points placed at the extremity 

 of the least and mean axes are thus represented by means of arcs of the 

 focal ellipse and hyperbola respectively. In consequence of the third focal 

 conic of the ellipsoid being imaginary, no direct geometrical representation can 

 be given for the attraction A on a point placed at the extremity of its greater 

 axis. It will, however, be found, as was intimated above, that a simple rela- 

 tion exists between the three attractions, which enables us to represent this 

 last by means of arcs of both focal conies. 



The relation alluded to is 



7+1 + 7 = ^'^^^''* ^'^ 



* Proceedings of the Royal Irish Academy, vol. ii. p. 525. 



