Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/265

the likeneſs of this orbit which a body acted upon by an uniform centripetal force deſcribes, and of that orbit which a body performing its circuits in a revolving ellipſis will deſcribe in a quieſcent plane. By this collation of the terms, theſe orbits are made ſimilar, not univerſally indeed, but then only when they approach very near to a circular fugure. A body therefore revolving with an uniform centripetal force in an orbit nearly circular, will always deſcribe an angle of $\textstyle {\frac {180}{\sqrt {3}}}$ deg. or 103 deg. 55 m. 23 ſec. at the centre; moving from the upper apſis to the lower apſis when it has once deſcribed that angle, and thence returning to the upper apſis when it has deſcribed that angle again; and ſo on in infinitum.

Exam. 2. Suppoſe the centripetal force to be as any power of the altitude A, as for example $\textstyle A^{n-3}$ or $\textstyle {\frac {A^{n}}{A^{3}}}$; where n - 3 and n ſignify any indices of powers whatever, whether integers or fractions, rational or ſurd, affirmative or negative. That numerator $\textstyle A^{n}$ or $\textstyle {\overline {T-X}}\vert ^{n}$ being reduced to an indeterminate ſeries by my method of converging ſeries, will become $\textstyle T^{n}-nXT^{n-2}+{\frac {nn-n}{2}}XXT^{n-2}$ &c. And conferring theſe terms with the terms of the other numerator RGG - RFF + TFF - FXX, it becomes as RGG - RFF + TFF to $\textstyle T^{n}$ ſo - FF to $\textstyle -nT^{n-1}+{\frac {nn-n}{2}}XT^{n-2}$ &c. And taking the laſt ratio's where the orbits approach to