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

 the likenes of this orbit which a body acted upon by an uniform centripetal force decribes, and of that orbit which a body performing its circuits in a revolving ellipis will decribe in a quiecent plane. By this collation of the terms, thee orbits are made imilar, not univerally 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 decribe an angle of $$\textstyle \frac {180}{\sqrt 3}$$ deg. or 103 deg. 55 m. 23 ec. at the centre; moving from the upper apis to the lower apis when it has once decribed that angle, and thence returning to the upper apis when it has decribed that angle again; and o on in infinitum.

Suppoe 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 ignify any indices of powers whatever, whether integers or fractions, rational or urd, affirmative or negative. That numerator $$\textstyle A^n$$ or $$\textstyle \overline{T - X} \vert ^n$$ being reduced to an indeterminate eries by my method of converging eries, will become $$\textstyle T^n - nXT^{n - 2} + \frac {nn - n}{2}XXT^{n - 2}$$ &c. And conferring thee terms with the terms of the other numerator RGG - RFF + TFF - FXX, it becomes as RGG - RFF + TFF to $$\textstyle T^n$$ o - FF to $$\textstyle - n T^{n - 1} + \frac {nn - n}{2}XT^{n - 2}$$ &c. And taking the lat ratio's where the orbits approach to