On the Attraction of Ellipsoids. 357 



K,Q! 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 



(a 2 - c 2 ) (b 2 - c 2 ) T ' 



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

 of the focal ellipse, such that 



OK" - / I ft 4- W 2 (h 2 *} \ 



c 



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

 OP be the perpendicular let fall from and KQ ; then i// denot- 

 ing the angle A OP, 



OK 2 . cos 2 iL = - - {c 2 + u 2 (b 2 -c 2 }}. cos 2 </,. 

 c 



Moreover, 



OK 2 . cos 2 ^ = OP 2 = (a- - c 2 ) cos 2 $ + (V - c 2 ) sin 2 $. 

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



sm > = 



~ rr sv 



c 2 + w 2 (a 2 - c 2 ) 

 Now 



d. (tan KQ, - arc A Q) = sin i///. OK* 



_ (a 2 - c 2 ) (6 2 - c 2 ) ___ u*du __ ___ 

 c " -/ {c 2 + M Z (a 2 - c 2 ) } -/ {c 2 + w 2 (b 2 - c 2 )' 



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 B + dQ, respectively, is 



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



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

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



