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 ame api; 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 $$\textstyle A \frac 14$$ or directly as $$\textstyle A^6$$ or $$\textstyle A^{13}$$. Latly, if the body in its progres from the upper apis to the ame upper apis again, goes over one entire revolution and three deg. more, and therefore that apis in each revolution of the body moves three deg. in conequentia; 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 decreaes in a ratio omething greater than the duplicate; but approaching 59$$\textstyle \frac 14$$ times nearer to the duplicate than the triplicate.

Hence alo if a body, urged by a centripetal force which is reciprocally as the quare of the altitude, revolves in an ellipis whoe focus is in the centre of the forces; and a new and foreign force hould be added to or ubducted from this centripetal force; the motion of the apides ariing from that foreign force may (by the third examples) be known; and o on the contrary. As if the force with which the body revolves in the ellipis be as $$\textstyle \frac 1{AA}$$; and the foreign