carried forwards. For the force with which the body P is urged to the body T at the quadratures, where the force MN vaniſhes, is compounded of the force LM and the centripetal force with which the body T attracts the body P. The firſt force LM, if the diſtance PT be increaſed. is increaſed in nearly the ſame proportion with that diſtance, and the other force decreaſes in the duplicate ratio of that diſtance; and therefore the ſum of theſe two forces decreaſe in a leſs than the duplicate ratio of the diſtance PT and therefore by cor. 1. prop. 45. will make the line of the apſides, or, which is the ſame thing, the upper apſis, to go backward. But at the conjunction and oppoſition the force with which the body P is urged towards the body T is the difference of the force KL, and of the force with which the body T attracts the body P; and that difference, becauſe the force KL is very nearly increaſed in the ratio of the diſtance PT; decreaſe in more than the duplicate ratio of the diſtance PT; and therefore by cor. 1. prop. 45. cauſes the line of the apſides to go forwards. In the places between the ſyzygies and the quadratures, the motion of the line of the apſides depends upon both theſe cauſes conjunctly, to that it either goes forwards or backwards in proportion to the exceſs of one of theſe cauſes above the other. Therefore ſince the force KL in the ſyzygies is almoſt twice as great as the force LM in the quadratures, the exceſs will be on the ſide of the force KL, and by conſequence the line of the apſides will be carried forwards. The truth of this and the foregoing corollary will be more eaſily underſtood by conceiving the ſyſtem of the two bodies T and P, to be ſurrounded on
Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/326
