Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/269

 eries whoe denominator is $$\scriptstyle A^3$$. Then the given part of the numerator ariing from that operation is to be uppoed in the ame 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 ame numerator which is not given. And taking away the uperfluous quantities and writing unity for T, the proportion of G to F is obtained

Hence if the centripetal force be as any power of the altitude. that power may be found from the motion of the apides; and o contrary-wie. That is, if the whole angular motion, with which the body returns to the ame apis, 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 $$\textstyle A \frac {nn}{mm} - 3$$ of the altitude A; the index of which power is $$\textstyle \frac {nn}{mm} - 3$$. This appears by the econd examples. Hence 'tis plain that the force in its reces from the centre cannot decreae in a greater than a triplicate ratio of the altitude. A body revolving with uch a force and parting from the apis, if it once begins to decend can never arrive at the lower apis or leat altitude, but will decend to the centre. decribing the curve line treated of in cor. 3. prop. 41. But if it hould, at its parting from the lower apis begin to acend never o little, it will acend in infinitum and never come to the upper apis; but will decribe the curve line poken of in the lame cot. and cor. 6. prop. 44. So that where the force in its reces from the centre descreaes