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



If the orbit VPK be an ellipis having its focus C, and its highet apis V, and we uppoe the ellipis upk imilar and equal to it, o 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 ellipis; the force with which a body may be made to revolve in a moveable ellipis will be as  $$\textstyle \frac {FF}{AA} + \frac {RGG - RFF}{A^3}$$  and vice vera. Let the force with which a body may revolve in an immovable ellipis, be expreed 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 ditance CV with the ame velocity as a body revolving in an ellipis has in V, is to the force with which a body revolving in an ellipis is acted upon in the apis V, as half the latus rectum of the ellipis, to the emi-diameter CV of the circle, and therefore is as $$\textstyle \frac {RFF}{CV^3}$$; and the force which is to thi as CG - FF to FF, is as $$\textstyle \frac {RCG - RFF}{CV^3}$$ (by of this prop.) is the difference of the forces in V with which the body P revolves in the immovable ellipis VPK, and the body p in the moveable ellipis upk, Therefore ince by this prop. that difference at any other altitude A is to it elf at the altitude CV as $$\textstyle \frac 1{A^3}$$ to $$\textstyle \frac 1{CV^3}$$, the