Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/304

 may be decribed in times that are in a ubduplicate ratio of the ditances; and the bodies P, p, always attracted by equal forces will decribe round the quiecent centres C and s imilar figures PQV, pqv, the latter of which pqv is imilar and equal to the figure which the body P decribes round the moveable body. Q. E. D.

Suppoe now that the common centre of gravity together with the pace in which the bodies are moved among themelves, proceeds uniformly in a right line; and (by cor. 6. of the laws of motion) all the motions in this pace will be performed in the ame manner as before; and therefore the bodies will decribe mutually about each other the ame figures as before, which will be thererefore imilar and equal to the figure pqv. Q. E. D.

Hence two bodies attracting each other with forces proportional to their ditance. decribe (by prop. 10.) both round their common centre of gravity, and round each other mutually, concentrical ellipes; and vice vera if uch figures are decribed, the forces are proportional to the ditances.

And two bodies, whoe forces are reciprocally proportional to the quare of their ditance decribe, (by prop. 11, 12, 13.) both round their common centre of gravity and round each other mutually, conic ections having their focus in the centre about which the figures are decribed And vice vera, if uch figures are decribed, the centripetal forces are reciprocally proportional to the quare of the ditance.

. Any two bodies revolving round their common centre of gravity, decribe areas proportional to