EXAM. 2. Suppose the centripetal force to be as any power of the altitude A, as, for example, , or ; where n - 3 and n signify any indices of powers whatever, whether integers or fractions, rational or surd, affirmative or negative. That numerator An or being reduced to an indeterminate series by my method of converging series, will become , &c. And conferring these terms with the terms of the other numerator RGG - RFF + TFF - FFX, it becomes as RGG - RFF + TFF to Tn, so -FF to , &c. And taking the last ratios where the orbits approach to circles, it becomes as RGG to Tn, so -FF to , or as GG to , so FF to ; and again, GG to FF, so to , that is, as 1 to n; and therefore G is to F, that is the angle VCp to the angle VCP, as 1 to . Therefore since the angle VCP, described in the descent of the body from the upper apsis to the lower apsis in an ellipsis, is of 180 deg., the angle VCp, described in the descent of the body from the upper apsis to the lower apsis in an orbit nearly circular which a body describes with a centripetal force proportional to the power , will be equal to an angle of deg., and this angle being repeated, the body will return from the lower to the upper apsis, and so on in infinitum. As if the centripetal force be as the distance of the body from the centre, that is, as or , n will be equal to 4, and equal to 2; and therefore the angle[Pg 179] between the upper and the lower apsis will be equal to deg., or 90 deg. Therefore the body having performed a fourth part of one revolution, will arrive at the lower apsis, and having performed another fourth part, will arrive at the upper apsis, and so on by turns in infinitum. This appears also from Prop. X. For a body acted on by this centripetal force will revolve in an immovable ellipsis, whose centre is the centre of force. If the centripetal force is reciprocally as the distance, that is, directly as or , n will be equal to 2; and therefore the angle between the upper and lower apsis will be deg., or 127 deg., 16 min., 45 sec.; and therefore a body revolving with such a force, will by a perpetual repetition of this angle, move alternately from the upper to the lower and from the lower to the upper apsis for ever. So, also, if the centripetal force be reciprocally as the biquadrate root of the eleventh power of the altitude, that is, reciprocally as , and, therefore, directly as , or as , n will be equal to , and deg. will be equal to 360 deg.; and therefore the body parting from the upper apsis, and from thence perpetually descending, will arrive at the lower apsis when it has completed one entire revolution; and thence ascending perpetually, when it has completed another entire revolution, it will arrive again at the upper apsis; and so alternately for ever.