Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/308



If two bodies S and P, attracting each other with forces reciprocally proportional to the quares of their ditance, revolve about their common centre of gravity; I ay that the principal axis of the ellipis which either of the bodies as P decribesr by this motion about the other S, will be to the principal axis of the ellipis, which the ame body P may decribe in the ame periodical time about the other body S quiecent, as the um of the two bodies S + P to the firt of two mean proportionals between that um and the other body S.

For if the ellipes decribed were equal to each other, their periodic times by the lat theorem would be in a ubduplicate ratio of the body S to the um of the bodies S + P. Let the periodic time in the latter ellipis be diminihed in that ratio, and the periodic times will become equal; but by prop. 15. the principal axis of the ellipis will be diminihed in a ratio equiplicate to the former ratio; that is in a ratio, to which the ratio of S to S + P is triplicate; and therefore that axis will be to the principal axis of the other ellipis. as the firt of two mean proportionals between