revolving at small distances from their Primaries. 411 



the hypothesis I have adopted ; and it would seem that in this, 

 as in many other problems of physical science, the cases which 

 occur in nature are such as the powers of analysis can reach with 

 the greatest facility. 



To begin with the case of a solid spherical satellite, which I 

 shall suppose to be capable of preserving its form unchanged by 

 the disturbance previous to the final rupture : let r denote its 

 radius, r 3 the radius of the primary, and g the measure of the 

 attraction exerted at the distance k by a portion of matter taken 

 as unity, and of equal density with the satellite, but immeasurably 

 inferior to it in magnitude. Put it for 3* 141 6; R for the radius 

 of the spherical space which the matter of the primary might 

 fill if reduced to the same density as the satellite ; and x for the 

 distance between the centres of both bodies. In the absence of 

 every disturbing influence, the force of gravity on the surface of 

 the satellite will be 



&|f. . a) 



The diminution of this gravity at the point nearest to the pri- 

 mary, by the attraction of the latter, is equal to 



4ffft 2 7rR 3 4ffft 2 7rR 3 

 3{x-r)* 3a 2 ' 



or 



4^VR 3 / n 3r 2 4T 3 5r 4 \ 



At 90 degrees from this point, gravity is augmented by the 

 quantity 



4^ 2 ttR 3 / 3r 3 15?- 5 \ 



Now if M and m denote the attractive force of both bodies at 

 the distance k, the period of the satellite's revolution (which I 

 shall call T) will be expressed by the formula 



T- 2 7r * f . (4) 



*VM + m ....... v ; 



As its time of rotation is also eq ual to T, the equatorial velocity 

 of the satellite is = kr !t3±S. Calling this v, the centrifugal 



X* 



V* 



force at the equator will be equal to — , or 



* 2 (M + m)r ' _ 



^ • • \P) 



2E2 



