OEC. IX.l OF NATURAL PHILOSOPHY. 175 revolving in an immovable orbit has by a radius drawn to the centre debribed in any certain time, the difference of the forces, with which the body P revolves in an immovable orbit, and the body p in a movable or bit, will be to the centripetal force, with which another body by a radius drawn to the centre can uniformly describe that sector in the same time as the area VPC is described, as GG — FF to FF. For that sector and the area pCk are to one another as the times in which they are described. COR. 2. If the orbit YPK be an ellipsis, having its focus C, and its highest apsis Y, and we suppose the the ellipsis upk similar and equal to .. it, so that pC may be always equal / to PC, and the angle YC/? be to the •; angle YCP in the given ratio of G \ to F ; and for the altitude PC or pC \ we put A, and 2R for the latus rec- /t\ turn of the ellipsis, the force with * which a body may be made to re volve in a movable ellipsis will be as FF RGG — RFF — - + - — -rg — — , and vice versa. /Y A. A. Let the force with which a body may revolve in an immovable ellipsis be expressed by the quantity , and the -. 7 force in V will be FF But the force with which a body may revolve in a circle at the distance CY, with the same velocity as a body revolving in an ellipsis has in Y, is to the force with which a body revolving in an ellip sis is acted upon in the apsis Y, as half the latus rectum of the ellipsis to the RFF semi-diameter CY of the circle, and therefore is as , =- : and tlu RFF which is to this, as GG — FF to FF, is as - ~py^~~ ~: and this force (by Cor. 1 cf this Prop.) is the difference of the forces in Y, with which the body P revolves in the immovable ellipsis YPK, and the body p in the movable ellipsis upk. Therefore since by this Prop, that difference at any other altitude A is to itself at the altitude CY as -r-, to ^TF-» the same AJ CYJ R C^ ( "* R P^ T*1 difference in every altitude A will be as - — 3 — : — . Therefore to the FF force -T-: , by which the body may revolve in an immovable ellipsis VPK