ſeries whoſe denominator is . 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 obtainecx
Cor. 1. Hence I the centripetal force be as any power of the altitude. that power may be found from the motion of the apſides; and ſo contrary-wiſe. 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 drg. as any number as m to another as n, and the altitude called A; the force will be as the power of the altitude A; the index of which power is . This appears by the ſecond examples. Hence 'tis 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. 41. 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 lame cot. and cor. 6. prop. 44. So that where the force in its receſs from the centre descreaſes
