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

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

. Let us uppoe the centripetal force to be uniform, and therefore as $$\textstyle \frac {A^3}{A^3}$$, or, writing T - X for A inthe numerator, as $$\textstyle \frac {T^3 - 3TTX + 3TXX - X^3}{A^3}$$ $$\textstyle \frac {T^3 - 3TTX + 3TXX - X^3}{A^3}$$. Then collating together the correpondent terms of the numerators, that is, thoe that confit of given quantities. with thoe of given quantities, and thoe of quantities not given, with thoe of quantities not given, it will become RGG - EFF + TFF to $$\scriptstyle T^3$$ as - FFX to $$- 3TTX + 3 TXX - X^3$$ or as - FF to - 3TT + 3TX - XX. Now ince the orbit is uppoed extreamly near to a circle, let it coincide with a circle, and becaue in that cae R and T become equal, and X is infinitely diminihed, the lat ratio's will be, as RGG to $$\textstyle T^3$$ o - FF to 3TT, or as GG to TT o FF to 3TT, and again as GG to FF o 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, $$\scriptstyle \sqrt 3$$. Therefore ince the body, in an immoveable ellipis, in decending from the upper to the lower apis, decribes an angle, if I may o peak, of 180 deg. the other body in a moveable ellipis, and therefore in the immovable orbit we are treating of, will, in its decent from the upper to the lower apis, decribe an angle VCp of $$\textstyle \frac {180}{\sqrt 3}$$ deg. And this comes to pa by reaon of