Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/271

or two thirds, or one third; or one fourth part of an entire revolution, return to the ſame apſiſ; m will be to n as $\scriptstyle 34$ or $\scriptstyle 23$ or $\scriptstyle 13$ or $\scriptstyle 14$ or to 1, and therefore $\textstyle A{\frac {nn}{mm}}-3$ is equal to $\textstyle A^{{\frac {16}{9}}-3}$ or $\textstyle A^{{\frac {9}{4}}-3}$, or $\textstyle A^{9-3}$, or $\textstyle A^{16-3}$; and therefore the force is either reciprocally as $\textstyle A^{\frac {11}{9}}$ or $\textstyle A{\frac {1}{4}}$ or directly as $\textstyle A^{6}$ or $\textstyle A^{13}$. Laſtly, if the body in its progreſs from the upper apſis to the ſame upper apſis again, goes over one entire revolution and three deg. more, and therefore that apſis in each revolution of the body moves three deg. in conſequentia; then m will be to n as 363 deg. to 360 deg. or as 121 to 120, and therefore $\textstyle A^{{\frac {nn}{mm}}-3}$ will be equal to $\textstyle A^{-{\frac {29523}{14641}}}$ and therefore the centripetal force will be reciprocally as $\textstyle A^{\frac {29523}{14641}}$ or reciprocally as $\textstyle A^{2{\frac {49}{253}}}$ very nearly. Therefore the centripetal force decreaſes in a ratio ſomething greater than the duplicate; but approaching 59$\textstyle {\frac {1}{4}}$ times nearer to the duplicate than the triplicate.

Cor. 1. Hence alſo if a body, urged by a centripetal force which is reciprocally as the ſquare of the altitude, revolves in an ellipſis whoſe focus is in the centre of the forces; and a new and foreign force ſhould be added to or ſubducted from this centripetal force; the motion of the apſides ariſing from that foreign force may (by the third examples) be known; and ſo on the contrary. As if the force with which the body revolves in the ellipſis be as $\textstyle {\frac {1}{AA}}$; and the foreign