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

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may be deſcribed in times that are in a ſubduplicate ratio of the diſtances; and the bodies P, p, always attracted by equal forces will deſcribe round the quieſcent centres C and s ſimilar figures PQV, pqv, the latter of which pqv is ſimilar and equal to the figure which the body P deſcribes round the moveable body. Q. E. D.

Case 2. Suppoſe now that the common centre of gravity together with the ſpace in which the bodies are moved among themſelves, proceeds uniformly in a right line; and (by cor. 6. of the laws of motion) all the motions in this ſpace will be performed in the ſame manner as before; and therefore the bodies will deſcribe mutually about each other the ſame figures as before, which will be thererefore ſimilar and equal to the figure pqv. Q. E. D.

Cor. 1. Hence two bodies attracting each other with forces proportional to their diſtance. deſcribe (by prop. 10.) both round their common centre of gravity, and round each other mutually, concentrical ellipſes; and vice verſa if ſuch figures are deſcribed, the forces are proportional to the diſtances.

Cor. 2. And two bodies, whoſe forces are reciprocally proportional to the ſquare of their diſtance deſcribe, (by prop. 11, 12, 13.) both round their common centre of gravity and round each other mutually, conic ſections having their focus in the centre about which the figures are deſcribed And vice verſa, if ſuch figures are deſcribed, the centripetal forces are reciprocally proportional to the ſquare of the diſtance.

Cor. 3.. Any two bodies revolving round their common centre of gravity, deſcribe areas proportional to