Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/260

Cor. 2. If the orbit VPK be an ellipſis having its focus C, and its higheſt apſis V, and we ſuppoſe the ellipſis upk ſimilar and equal to it, ſo that pC may be always equal to PC, and the angle VCp be to the angle VCP in the given ratio of G to F; and for the altitude PC or pC we put A, and a R for the latus rectum of the ellipſis; the force with which a body may be made to revolve in a moveable ellipſis will be as $\textstyle {\frac {FF}{AA}}+{\frac {RGG-RFF}{A^{3}}}$ and vice verſa. Let the force with which a body may revolve in an immovable ellipſis, be expreſſed by the quantity $\textstyle {\frac {FF}{AA}}$, and the force in V will be $\textstyle {\frac {FF}{CV^{2}}}$. But the force with which a body may revolve in a circle at the diſtance CV with the ſame velocity as a body revolving in an ellipſis has in V, is to the force with which a body revolving in an ellipſis is acted upon in the apſis V, as half the latus rectum of the ellipſis, to the ſemi-diameter CV of the circle, and therefore is as $\textstyle {\frac {RFF}{CV^{3}}}$; and the force which is to thiſ as CG - FF to FF, is as $\textstyle {\frac {RCG-RFF}{CV^{3}}}$ (by cor. 1. of this prop.) is the difference of the forces in V with which the body P revolves in the immovable ellipſis VPK, and the body p in the moveable ellipſis upk, Therefore ſince by this prop. that difference at any other altitude A is to it ſelf at the altitude CV as $\textstyle {\frac {1}{A^{3}}}$ to $\textstyle {\frac {1}{CV^{3}}}$, the