494 Mr. D. Vaughan on Static and Dynamic Stability 



in which h denotes the radius of the primary supposed to be a 

 sphere, D its distance, and n its density divided by that of the 

 satellite. In the article just referred to, it has been shown that 

 P', Q', and R' must be reciprocally proportional to A, B, and C ; 

 and accordingly by equalling the values of AP', BQ', and CR' as 

 deduced from the last equation, we obtain 



.... (16) 



whence 



6 - D 3 > € l - 2)3 



e 2 : A-C : /B-C 



■?=-, and 



.4^5). . . m 



-i 4 . A 



If the satellite were not homogeneous, the ratio between the 

 greatest and least ellipticities would vary between 4 and 6, the 

 latter number expressing the ratio in the case in which the cen- 

 tral matter alone is supposed to be endued with attractive power. 

 The extent to which the form of the satellite affects the inten- 

 sity of the force which binds it to the primary, supposing this 

 body to be a sphere, may be readily found by means of Ivory's 

 theorem ; and the application will be facilitated in the present 

 case, in which the external point ranges with the axis A. The 

 effect of the attractive force in moving the primary will be 

 47r%ABC / 3 A 2 e 2 3 A«e»\ 



3D 2 V ^5 D 2 10 W )' ' ' y ' 

 But the same amount of matter in a spherical form would attract 



the central orb with a force expressed by \~> ; so that 



the excess of attractive power due to the ellipticity is 



%^(!^-^v). . . (19 , 



Calling this F, and putting m for 4 ^ ABC , and t f or e , 2 , in 



o 4 



accordance with formula (17), 



t, 21 mA 2 e 2 



F= 40-JF- ( 2 °) 



To show the effects of the change of form in consequence of 

 the eccentricity of the orbit, which is to be regarded as deviating 

 little from a circle, wc must take the variation of the last formula. 

 Then 



21mAe/ AC , . ,. 2AeSD 

 20 D 4 



ASe + eSA ^— J. . . (21) 



/ 1 — e 2 \ 



But the volume of the ellipsoid is equal to A 3 f ) or 



