Page:Newton's Principia (1846).djvu/181

 revolving in an immovable orbit has by a radius drawn to the centre described 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 orbit, 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.

. 2. If the orbit VPK be an ellipsis, having its focus C, and its highest apsis V, and we suppose the 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 2R for the latus rectum of the ellipsis, the force with which a body may be made to revolve in a movable ellipsis will be as $$\scriptstyle \frac{FF}{AA}+\frac{RGG-RFF}{A^{3}}$$, and vice versa. Let the force with which a body may revolve in an immovable ellipsis be expressed by the quantity$$\scriptstyle \frac{FF}{AA}$$, and the force in V will be $$\scriptstyle \frac{FF}{CV^{2}}$$. 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 $$\scriptstyle \frac{RFF}{CV^{3}}$$; and the force which is to this, as GG - FF to FF, is as $$\scriptstyle \frac{RGG-RFF}{CV^{3}}$$: and this force (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 movable ellipsis upk. Therefore since by this Prop, that difference at any other altitude A is to itself at the altitude CV as $$\scriptstyle \frac{1}{A^{3}}$$to $$\scriptstyle \frac{1}{CV^{3}}$$, the same difference in every altitude A will be as $$\scriptstyle \frac{RGG-RFF}{A^{3}}$$. Therefore to the force $$\scriptstyle \frac{FF}{AA}$$, by which the body may revolve in an immovable ellipsis VPK