Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/267

therefore the angle between the upper and lower apſis will be of deg. or 127 deg. 16 min. 45 ſec. and therefore a body revolving with ſuch a force, will, by a perpetual repetition of this angle, move alternately from the upper to the lower, and from the lower to the upper apſis for ever. So alſo if the centripetal force be reciprocally as the biquadrate root of the eleventh power of the altitude, that is reciprocally as $\textstyle A{\frac {11}{4}}$ and therefore directly as $\textstyle {\frac {1}{A^{\frac {11}{4}}}}$ or as $\textstyle {\frac {A^{\frac {3}{4}}}{A^{3}}}$, as will be equal to $\textstyle {\frac {1}{4}}$ and $\textstyle {180}{\sqrt {n}}$ deg. will be equal to 360 deg. and therefore the body parting from the upper apſis, and from thence perpetually deſcending will arrive at the lower apſis when it has compleated one entire revolution; and thence aſcending perpetually, when it has compleated another entire revolution it will arrive again as the upper apſis; and ſo alternately for ever.

Exam. 3.. Taking m and n for any indices of the powers of the altitude, and b and c for any given numbers, ſuppoſe the centripetal force to be as $\textstyle {\frac {b\ into\ {\overline {T-X}}\vert ^{m}c\ into\ {\overline {T-X}}\vert ^{n}}{A^{3}}}$ or (by the method of converging ſeries above-mentioned) as $\textstyle {\frac {bT^{m}+cT^{n}-mbXT^{m-1}ncXT^{n-2}}{A^{3}}}+{\frac {mm-m}{2}}bXXT^{m-2}+{\frac {nn-n}{2}}cXXT^{n-2}$ &c. and comparing the terms of the numerators,