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

 than LM is more diminihed than increaed; and moreover ince that centripetal force (by cor. 1. prop. 4.) is in a ratio compounded of the imple ratio of the radius TP directly, and the duplicate ratio of the periodical time inverely; it is plain that this compounded ratio is diminihed by the action of the force KL; and therefore that the periodical time, uppoing the radius of the orbit PT to remain the ame, will be increaed. and that in the ubduplicate of that ratio in which the centripetal force is diminihed and therefore uppoing this radius increaed or diminihed, the periodical time will be increaed more or diminihed les than in the equiplicate ratio of this radius, by cor. 6. prop. 4. If that force of the central body hould gradually decay, the body P being les an les attracted would go farther and farther from the centre T; and on the contrary if it were increaed it would draw nearer to it. Therefore if the action of the ditant body S, by which that force is diminihed, were to increae and decreae by turns; the radius TP will be alo increaed and diminihed by turns; and the periodical time will be increaed and diminihed in a ratio compounded of the equiplicate ratio of the radius, and of the ubduplicate of that ratio in which the centripetal force of the central body T is diminihed or increaed by the increae or decreae of the action of the ditant body S.

It alo follows from what was before aid down, that the axis of the ellipis decribed by the body P, or the line of the apides, does as to its angular motion go forwards and backwards by turns, but more forwards than backwards, and by the exces of its direct motion, is in the whole