Page:Newton's Principia (1846).djvu/183

body will deflect from the rectilinear motion into the curve line Vpk. But this curve Vpk is the same with the curve VPQ found in Cor. 3, Prop XLI, in which, I said, bodies attracted with such forces would ascend obliquely.

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 movable 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 any 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 $\scriptstyle {\frac {FF}{AA}}+{\frac {RGG-RFF}{A^{3}}}$, that is as, $\scriptstyle {\frac {FFA+RGG-RFF}{A^{3}}}$, by substituting T - X for A; will become as $\scriptstyle {\frac {RGG-RFF+TFF-FFX}{A^{3}}}$. In like manner any other centripetal force is to be reduced to a fraction whose denominator is A³, and the numerators are to be made analogous by collating together the homologous terms. This will be made plainer by Examples.

Example 1. Let us suppose the centripetal force to be uniform, and therefore as $\scriptstyle {\frac {A^{3}}{A^{3}}}$ or, writing T - X for A in the numerator, as $\scriptstyle {\frac {T^{3}-3TTX+3TXX-X^{3}}{A^{3}}}$. Then collating together the correspondent terms of the numerators, that is, those that consist of given quantities, with those of given quantities, and those of quantities not given with those of quantities not given, it will become RGG - RFF + TFF to T³ as - FFX to 3TTX + 3TXX - X³, or as - FF to - 3TT + 3TX - XX. Now since the orbit is supposed extremely near to a circle, let it coincide with a circle; and because in that case R and T become equal, and X is infinitely diminished, the last ratios will be, as RGG to T², so - FF to - 3TT, or as GG to TT, so FF to 3TT; and again, as GG to FF, so TT to 3TT, that is, as 1 to 3; and therefore G is to F, that is, the angle VCp to the angle VCP, as 1 to $\scriptstyle {\sqrt {3}}$. Therefore since the body, in an immovable