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



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

This problem is olved arithmetically by reducing the orbit, which a body revolving in a moveable ellipis (as in cor. 2. and 3 of the above prop.) decribes in an immovable plane, to the figure of the orbit whoe apides are required; and then eeking the apides of the orbit which that body decribes in an immovable plane. But orbits acquire the ame figure, if the centripetal forces with which they are decribed, compared between themelves, are made proportional at equal altitudes. Let the point V be the highet apis, and write T for the greatet 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 ellipis 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 ubstituting 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 whoe denominator is A and the numerators are to be made analogous by