On the Attraction of Ellipsoids. 357 



KiQi to the focal ellipse, and let T = tan KiQ! - arc A Qi ; then 

 the attraction of the ellipsoid on the particle /* placed at the ex- 

 tremity C of the least axis is 



4tr fjifp abc* 



(a 2 - c 2 ) (b* - c*) 



T. (2) 



For let a point K (see fig. 3) be assumed on the greater axis OA 

 of the focal ellipse, such that 



OA 



from K let a tangent KQ, be drawn to the focal ellipse, and let 

 OP be the perpendicular let fall from and KQ, ; then ^ denot- 

 ing the angle A OP, 



OK 2 , cos 2 1/, = ^- {c 2 + w 2 (V - c 2 ) J . cos 2 </,. 

 c 



Moreover, 



OK 2 , cos 2 ^ = OP 2 = (<# - c 2 ) cos 2 ^ + (b z - c 2 ) sin 2 ^. 

 Equating these values, and solving for sin 2 *//, we get 



(a* - ") u z 



V 



Sin I/ = 



. . . -- rr-. 



c 2 + u? (a 2 - c 2 ) 

 Now 



d. (tan KQ - arc A Q) = sin $d. OK 



(a* - c 2 ) (b* - c 2 ) u 2 du 



By comparing this expression with (1), given in the last propo- 

 sition, it appears that the attraction on the point C of the portion 

 of the ellipsoid contained between the two conical surfaces whose 

 semi- angles are and 9 + dO, respectively, is 



* Transactions of the Royal Irish Academy, VOL. xvi. p. 79. Proceedings of 

 the Royal Irish Academy, VOL. n. p. 507 (supra, p. 255). 



