Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/263

Proposition XLV. Problem XXXI.

To find the motion of the apſides in orbits approaching very near to circles.

This problem is ſolved arithmetically by reducing the orbit, which a body revolving in a moveable ellipſis (as in cor. 2. and 3 of the above prop.) deſcribes in an immovable plane, to the figure of the orbit whoſe apſides are required; and then ſeeking the apſides of the orbit which that body deſcribes in an immovable plane. But orbits acquire the ſame figure, if the centripetal forces with which they are deſcribed, compared between themſelves, are made proportional at equal altitudes. Let the point V be the higheſt apſis, and write T for the greateſt altitude CV, A for an other altitude CP or Cp, and X for the difference of the altitudes CV - CP, and the force with which a body moves in an ellipſis revolving about its focus C (as in cor. 2.) and which in cor. 2. was as $\textstyle {\frac {FF}{AA}}+{\frac {RGG-RFF}{A^{3}}}$, that is as $\textstyle {\frac {FAA+RGG-RFF}{A^{3}}}$, by ſubstituting T - X for A will become $\textstyle {\frac {RGG-RFF+TFF-FFX}{A^{3}}}$. In like manner any other centripetal force is to be reduced to a fraction whoſe denominator is A and the numerators are to be made analogous by