in the Secondary Systems. 493 



to form and density, yet we may more easily arrive at definite 

 results by taking, as the most appropriate type of the figure of 

 these bodies, the ellipsoid which a homogeneous fluid satellite 

 must assume when its motions are adapted for keeping the 

 same point of its surface always directed to the centre of the 

 primary. Let A, B, and C represent the semiaxes ; P, Q, and 

 11 the attractions at their extremities in the absence of all dis- 



£2 Q2 JJ2 Q2 



turbing influences ; and put — p — = e 2 and — ™ — = e / 2 « 



By a course of investigation similar to that which I adopted for 

 finding the attraction of a prolate spheroid in the Philosophieal 

 Magazine (vol. xx. p. 414) the following result may be ob- 

 tained : — 



3ffA 2 C 2 CC cos 2 <ft sin 



rk*c*rr ( 



A JJl- 2 



cos^-efsin^cos 2 ^ ' ^ 

 in which g denotes the attractive force at the distance k of a 

 small portion of the body, <ft the angle formed with the axis A 

 by any of the elementary pyramids extending from its extremity 

 to the surface of the ellipsoid, and 6 the angle which the projec- 

 tion of these pyramids on the plane of B and C forms with B. 

 A double integration by series, rejecting the fourth and higher 

 powers of e and e„ gives 



p= ^ (l+ a^ + ^. . . . (]0) 



In like manner, by a slight modification of the process employed 

 in the same article (page 415) for finding the attraction at the 

 extremity of the minor axis of the prolate spheroid, we may 



^4^0^ | + ^ _ _ _ (n) 



K=«(l + f^). • • • <»> 



Let P', Q', and II' represent the actual intensity of gravity at 

 the extremity of each axis, taking into consideration the effects 

 of centrifugal force and the disturbance of the primary, to which 

 the axis A is always directed, while C is perpendicular to the 

 plane of the orbit ; for this condition is necessary for the equili- 

 brium, as I have shown in the Philosophical Magazine for April 

 1861. Then 



P= «( 1+ ^ + £)_^^ . (13) 



Q , = W( 1 + ! + 3£), (1 ,) 



w- 4^( 1+ £ + ^^jMF0 , . (15) 



