﻿516 Mr. D. Vaughan on Secondary Planets 



putting the other ordinates successively equal to nothing, there 



is obtained from (17), 



MFC 2 

 BQ=S, and RC+^r i =S=BQ; . . (19) 



whence 

 and 



.,. 3M* 2 A 2 Rr± MW 



PA-RC=^(3A 2 + C 2 ). .... (20) 



Putting for M its value -^^, r being the radius of the primary 



o 



supposed to be a sphere, and g the attractive force of a unit of 

 matter at the distance k, the last equation becomes 



PA~RC=^x^^(3A 2 + C 2 ) (21) 



The ellipticity at a given distance from the primary may be 

 found by the foregoing equations ; but my main object is to de- 

 termine the greatest ellipticity which the body could exhibit, and 

 the radius of the smallest circular orbit in which it could main- 

 tain its integrity. ISow equilibrium ceases to be possible on 

 the surface of the satellite, not when the disturbance of the cen- 

 tral orb wholly neutralizes gravity in the direction of the axis A, 

 but when it becomes too great to serve as a counterpoise to the 

 forces acting in the direction of A and C. This disturbance is 



proportional to ^ ; and accordingly the maximum value which, 



for different degrees of elongation, must be given by the last 



equation to ^r, will enable us to express, in terms of the radius 



of the primary, the radius of the smallest circle which the 

 satellite can describe. In the Philosophical Magazine for De- 

 cember 1860 I made, on this principle, an approximate estimate 

 of the size of the smallest orbit, regarding the satellite as a pro- 

 late spheroid ; but for strict accuracy in calculating this item, and 

 also the ratio of the principal axes of the body, it is necessary to 

 have recourse to elliptic integrals. 



For the sake of brevity, I shall avail myself of the results of 

 the investigation which Legendre has given for the attraction of 

 a homogeneous ellipsoid, in his Eocercices de Calcul Integral, 

 vol. ii. p. 523. His formulae, with some modifications in the no- 

 tation, and taking the density as unity, give 



