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

 circles, it becomes as RGG to $$\textstyle T^n$$ o -FF to $$\textstyle - nT^{n - 1}$$, or as GG to $$\textstyle - T^{n - 1}$$ o FF to $$\textstyle nT^{n - 1}$$, and again GG to FF o $$\textstyle T^{n - 1}$$ 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 ince the angle VCP, decribed in the decent of the body from the upper apis to the lower apis in an ellipis, is of 180 deg. the angle VCp, decribed in the decent of the body from the upper apis to the lower apis in an orbit nearly circular which a body decribes 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 apis, and o on in infinitum. As if the centripetal force be as the ditance 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 apis 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 apis, and having performed another fourth part, will arrive at the upper apis, and o on by turns in infinitum. This appears alo from prop. 10. For a body acted on by this centripetal force will revolve in an immovable ellipis, whoe centre is the centre of force. If the centripetal force is reciprocally as the ditance, that is, directly as $$\scriptstyle \frac 1A$$ or as $$\scriptstyle \frac {A^2}{A^3}$$, n will be equal to 2, and