304 PHILOSOPHICAL TRANSACTIONS. [aNNO 1758. 



centre of the primary planet, is to the satellite's gravity to its primary, as 



■^ -^ to 1 ; or, because by prop. 2, j is = m X sh (fig. Q), viz. putting m 



for the sine of the inclination of the planet's orbit to the equator of the primary, 



3bc 



and writing y for sh, as -t^ X (l — Sm^y*) to 1 ; and the sum of these forces in 

 the whole circumference whose radius is 1, is to the gravity of the satellite as 

 often taken, as -^ X (l — -^m^) to 1. Therefore the mean force, which maybe 

 supposed to act uniformly on the satellite, while it makes its period in an orbit nearly 



3hc 



circular, is to its gravity to the primary, as ^ X (1 — ^rn^) to 1 ; and l3y this 

 force will the apses be moved, if no regard be had to the other force perpendi- 

 cular to the radius of the orbit, and which through one-half of the satellite's 

 revolution tends one w^ay, and the contrary way through the other half Now 

 because, from what has been demonstrated in this and prop. 1, it follows that 

 the gravity of the satellite about the planet, whose figure is an oblate spheroid, 

 revolving generally at the distance /, is to the gravity of the same at the greater 

 distance l, as ^ -|- ^ x (l — ^ni^) to ~ + p x (l — ^m'^), b being a given 



11b b 



quantity of very small value, or as ^ to - — ^ (l — 4m') -\ ^ (l — 4m^) 



very nearly ; therefore the gravity of the satellite is diminished more than in the 

 duplicate ratio of the distance increased when m is less than V^, that is, when 

 the inclination of the satellite's orbit to the planet's equator is less than 54° 44'; 

 but is diminished in a less ratio when m is greater than -vZ-f-^ that is, when that 

 inclination exceeds 54° 44"; and therefore in the former case the apses of the 

 satellite's orbit proceeds forward, in the latter recedes. And the quantity of this 

 progress or regress will be thus known. 



By exam. 3, prop. 45, lib. 1, Newton's Princip. if to the centripetal force, 



1 c • 



which is as ^, be added another force as ^ that is, which may be to the centri- 

 fugal force ^, as .- to 1, the angle of revolution from the one apsis to the same 



\ A- t 360° 



will be 360° VyZT'* ^^YZT ^^^y "^^^^y* ^^ quantity e being very small. Fur- 

 ther, since the motion of the satellite revolving in its orbit, is to the motion of 



360" 360*^ 

 the apsis, as — — to —- 36o°, that is, as 1 to e, the motion of the apsis, 



in the time of the satellite's revolution to a star, will be = 36o° x e\ and this 

 motion of the apsis will be to the motion of the same in any other given time, 

 as the periodic time of the satellite is to that given time. But in our prop, e = 



^ (1 — 47»^); hence is given the motion of the apsis sought, q. e. i. 



CoROL. If this determination be referred to the moon, there will be Z? = 1 , 

 / = 60, m = sine of 23° 28^'; and if c = -^^ ; then will e = -mr^-nnrj and 



