Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/266

circles, it becomes as RGG to $\textstyle T^{n}$ ſo -FF to $\textstyle -nT^{n-1}$, or as GG to $\textstyle -T^{n-1}$ ſo FF to $\textstyle nT^{n-1}$, and again GG to FF ſo $\textstyle T^{n-1}$ to $\textstyle nT^{n-1}$, that is, as 1 to n; and therefore G is to F, that is the angle VCp to the angle VCP as 1 to $\scriptstyle {\sqrt {n}}$. Therefore ſince the angle VCP, deſcribed in the deſcent of the body from the upper apſis to the lower apſis in an ellipſis, is of 180 deg. the angle VCp, deſcribed in the deſcent of the body from the upper apſis to the lower apſis in an orbit nearly circular which a body deſcribes with a centripetal force proportional to the power $\textstyle A^{n-3}$, will be equal to an angle of $\textstyle {\frac {180}{\sqrt {n}}}$ deg. and this angle being repeated the body will return from the lower to the upper apſis, and ſo on in infinitum. As if the centripetal force be as the diſtance of the body from the centre, that is, as A, or $\textstyle {\frac {A^{n}}{A^{3}}}$, n will be equal to 4 and, $\scriptstyle {\sqrt {n}}$ equal to 2; and therefore the angle between the upper and the lower apſis will be equal to $\textstyle {\frac {180}{2}}$ deg. or 90 deg. Therefore the body having performed a fourth part of one revolution will arrive at the lower apſis, and having performed another fourth part, will arrive at the upper apſis, and ſo on by turns in infinitum. This appears alſo from prop. 10. For a body acted on by this centripetal force will revolve in an immovable ellipſis, whoſe centre is the centre of force. If the centripetal force is reciprocally as the diſtance, that is, directly as $\scriptstyle {\frac {1}{A}}$ or as $\scriptstyle {\frac {A^{2}}{A^{3}}}$, n will be equal to 2, and