Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/264

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collating together the homologous terms. This will be made plainer by examples.

Exam 1.. Let us ſuppoſe the centripetal force to be uniform, and therefore as , or, writing T - X for A inthe numerator, as . Then collating together the correſpondent terms of the numerators, that is, thoſe that confiſt of given quantities. with thoſe of given quantities, and thoſe of quantities not given, with thoſe of quantities not given, it will become RGG - EFF + TFF to as - FFX to or as - FF to - 3TT + 3TX - XX. Now ſince the orbit is ſuppoſed extreamly near to a circle, let it coincide with a circle, and becauſe in that caſe R and T become equal, and X is infinitely diminiſhed, the laſt ratio's will be, as RGG to ſo - FF to 3TT, or as GG to TT ſo FF to 3TT, and again as GG to FF ſo TT to 3TT, that is, as 1 to 3; and therefore G is to F, that is, the angle VCp to the angle VCP as 1 to, . Therefore ſince the body, in an immoveable ellipſis, in deſcending from the upper to the lower apſis, deſcribes an angle, if I may ſo ſpeak, of 180 deg. the other body in a moveable ellipſis, and therefore in the immovable orbit we are treating of, will, in its deſcent from the upper to the lower apſis, deſcribe an angle VCp of deg. And this comes to paſ by reaſon of