Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/261

ſame difference in every altitude A will be as $\textstyle {\frac {RGG-RFF}{A^{3}}}$. Therefore to the force $\textstyle {\frac {FF}{AA}}$, by which the body may revolve in an immovable ellipſis VPK, add the exceſs $\textstyle {\frac {RGG-RFF}{A^{3}}}$ and the ſum will be the whole force $\textstyle {\frac {FF}{AA}}+{\frac {RGG-RFF}{A^{3}}}$ by which a body may revolve in the ſame time in the moveable ellipſis upk.

Cor. 3. In the ſame manner it will be found that if the immovable orbit VPK be an ellipſis having its centre in the centre of the forces C; and there be ſuppoſed a moveable ellipſis upkk ſimilar, equal, and concentrical to it; and 2 R be the principal latus reclum of that ellipſis, and 2 T the latus tranſverſum or greater axis; and the angle VCp be continually to the angle VCP as G to F; the forces with which bodies may revolve in the immovable and moveable ellipſis in equal times, will be as $\textstyle {\frac {FF}{T^{3}}}$ and $\textstyle {\frac {FFA}{T^{3}}}+{\frac {RGG-RFF}{A^{3}}}$ reſperctively.

Cor. 4. And univerſally, if the greateſt altitude CV of the body be called T, and the radius of the curvature which the orbit VPK has in V, that is, the radius of a circle equally curve, be called R, and the centripetal force with which a body may revolve in any immovable trajectory VPK at the place K be called $\textstyle {\frac {VFF}{TT}}$, and in other places P be indefinitely stiled X; and the altitude CP be called