EXAM. 3. Taking m and n for any indices of the powers of the altitude, and b and c for any given numbers, suppose the centripetal force to be as , that is, as or (by the method of converging series above-mentioned) as , &c. and comparing the terms of the numerators, there will arise RGG - RFF + TFF to bTm + cTn as -FF to -mbTm - 1 - ncTn - 1 , &c. And taking the last ratios that arise when the orbits come to a circular form, there will come forth GG to bTm - 1 + cTn - 1 as FF to mbTm - 1 + ncTn - 1; and again, GG to FF as bTm - 1 + cTn - 1 to mbTn - 1 + ncTn - 1. This proportion, by expressing the greatest altitude CV or T arithmetically by unity, becomes, GG to FF as b + c to mb + nc, and therefore as 1[Pg 180] to . Whence G becomes to F, that is, the angle VCp to the angle VCP, as 1 to . And therefore since the angle VCP between the upper and the lower apsis, in an immovable ellipsis, is of 180 deg., the angle VCp between the same apsides in an orbit which a body describes with a centripetal force, that is, as , will be equal to an angle of deg. And by the same reasoning, if the centripetal force be as , the angle between the apsides will be found equal to deg. After the same manner the Problem is solved in more difficult cases. The quantity to which the centripetal force is proportional must always be resolved into a converging series whose denominator is A3. Then the given part of the numerator arising from that operation is to be supposed in the same ratio to that part of it which is not given, as the given part of this numerator RGG - RFF + TFF - FFX is to that part of the same numerator which is not given. And taking away the superfluous quantities, and writing unity for T, the proportion of G to F is obtained.