SEC. IX..J OF NATURAL PHILOSOPHY. 177 body will deflect from the rectilinear motion into the curve line Ypk. But this curve ~Vpk is the same with the curve VPQ found in Cor. 3, Prop XLI, in which, I said, hodies attracted with such forces would ascend obliquely. PROPOSITION XLV. PROBLEM XXXL 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 C/?, and X for the difference of the altitudes C V — CP : and the force writh which a body moves in an ellipsis revolving about its •p -p T? C* f ^ T? F* F focus C (as in Cor. 2), and which in Cor. 2 was as -r-r -\ — — -.-3 — — , FFA + RGG — RFF , that is as, -^ — , by substituting T — X for A, will beA RGG — RFF + TFF — FFX come as -p — . In like manner any other cen tripetal force is to be reduced to a fraction whose denominator is A3, and the numerators are to be made analogous by collating together the homo logous terms. This will be made plainer by Examples. EXAMPLE 1. Let us suppose the centripetal force to be uniform, A3 and therefore as —3 or, writing T — X for A in the numerator, as T3 — 3TTX + 3TXX — X3 — — — =-. Ihen collating together the corresponA3 dent 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 -f- TFF to T3 as — FFX to 3TTX -f 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 VC/? to the angle VCP, as 1 to v/3. Therefore since the body, in an immovable