89 



of them, must be inversely as the cube of their common 

 distance from the centre, or of the altitude of the revolv 

 ing body in its orbit. Hence it follows that d being the 

 common altitude or distance, and P the parameter, the 

 force required to move the body in the moveable ellipse 



O T^l / 9 O\ 



m 2 r(n* m z ) , , . , . , _ 



is as -j2 x pTa 9 anc ^ conversely, if such is the force, 



the motion will be in a moveable ellipse : And again, 

 if a be the transverse axis of the ellipse, the forces in 

 the fixed and in the moveable orbit will be to each other as 



nf d . 



=- and 



.p. (ft 2 m 2 ) TT 



&quot; . ,, -. Hence, m order that 



a body may move in a moveable ellipse, or an arc which 

 advances or moves round in the direction of the body s 

 motion, the centripetal force must vary in a higher propor 

 tion than the inverse square of the distance, but less than 

 the cube ; and that the body may move in a retirincr 

 ellipse, or an arc which moves round in a direction 

 contrary to that of the body s motion, the centripetal force 

 must vary in a less proportion than the inverse square of the 

 distance. 



From these propositions, Sir Isaac Newton is enabled 



to ascertain the proportion of the centripetal force to 



the distance, when the motion of the elliptical axis, that 



_is^of_ the apsides, or extreme points of it, shall be given ; 



and conversely~&quot;to ascertain the motion of the apsides 

 when the proportion of the centripetal force to the dis 

 tance is given. Let &amp;lt;p : be the proportion of the angular 

 motion by which the body in the moveable orbit comes 

 round to the same lines of apsides, to the angular motion 

 of one revolution, or 360; then the centripetal force will 

 be as the power of the distance d, which is represented 



by ?5 3. Thus, if &amp;lt;p = 6, or the axis of the move- 



