Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/325

This page has been proofread, but needs to be validated.

than LM is more diminiſhed than increaſed; and moreover ſince that centripetal force (by cor. 1. prop. 4.) is in a ratio compounded of the ſimple ratio of the radius TP directly, and the duplicate ratio of the periodical time inverſely; it is plain that this compounded ratio is diminiſhed by the action of the force KL; and therefore that the periodical time, ſuppoſing the radius of the orbit PT to remain the ſame, will be increaſed. and that in the ſubduplicate of that ratio in which the centripetal force is diminiſhed and therefore ſuppoſing this radius increaſed or diminiſhed, the periodical time will be increaſed more or diminiſhed leſs than in the ſeſquiplicate ratio of this radius, by cor. 6. prop. 4. If that force of the central body ſhould gradually decay, the body P being leſs an leſs attracted would go farther and farther from the centre T; and on the contrary if it were increaſed it would draw nearer to it. Therefore if the action of the diſtant body S, by which that force is diminiſhed, were to increaſe and decreaſe by turns; the radius TP will be alſo increaſed and diminiſhed by turns; and the periodical time will be increaſed and diminiſhed in a ratio compounded of the ſeſquiplicate ratio of the radius, and of the ſubduplicate of that ratio in which the centripetal force of the central body T is diminiſhed or increaſed by the increaſe or decreaſe of the action of the diſtant body S.

Cor. 7. It alſo follows from what was before ſaid down, that the axis of the ellipſis deſcribed by the body P, or the line of the apſides, does as to its angular motion go forwards and backwards by turns, but more forwards than backwards, and by the exceſs of its direct motion, is in the whole