Page:Newton's Principia (1846).djvu/186

to $\scriptstyle {\frac {mb+nc}{b+c}}$. Whence G becomes to F, that is, the angle VCp to the angle VCP, as 1 to $\scriptstyle {\sqrt {\frac {mb+nc}{b+c}}}$. 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 $\scriptstyle {\frac {bA^{m}+cA^{n}}{A^{3}}}$, will be equal to an angle of $\scriptstyle 180{\sqrt {\frac {b+c}{mb+nc}}}$ deg. And by the same reasoning, if the centripetal force be as $\scriptstyle {\frac {bA^{m}-cA^{n}}{A^{3}}}$, the angle between the apsides will be found equal to $\scriptstyle 180{\sqrt {\frac {b-c}{mb-nc}}}$. 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.

Cor. 1 . Hence if the centripetal force be as any power of the altitude, that power may be found from the motion of the apsides; and so contrariwise. That is, if the whole angular motion, with which the body returns to the same apsis, be to the angular motion of one revolution, or 360 deg., as any number as m to another as n, and the altitude called A; the force will be as the power $\scriptstyle A^{{\frac {nn}{mm}}-3}$ of the altitude A; the index of which power is $\scriptstyle {\frac {nn}{mm}}-3$. This appears by the second example. Hence it is plain that the force in its recess from the centre cannot decrease in a greater than a triplicate ratio of the altitude. A body revolving with such a force and parting from the apsis, if it once begins to descend, can never arrive at the lower apsis or least altitude, but will descend to the centre, describing the curve line treated of in Cor. 3, Prop. XLI. But if it should, at its parting from the lower apsis, begin to ascend never so little, it will ascend in infinitum, and never come to the upper apsis; but will describe the curve line spoken of in the same Cor., and Cor. 6; Prop. XLIV. So that where the force in its recess from the centre decreases in a greater than a triplicate ratio of the altitude, the body at its parting from the apsis, will either descend to the centre, or ascend in infinitum, according as it descends or ascends at the beginning of its motion. But if the force in its recess from