Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/326

 carried forwards. For the force with which the body P is urged to the body T at the quadratures, where the force MN vanihes, is compounded of the force LM and the centripetal force with which the body T attracts the body P. The firt force LM, if the ditance PT be increaed. is increaed in nearly the ame proportion with that ditance, and the other force decreaes in the duplicate ratio of that ditance; and therefore the um of thee two forces decreae in a les than the duplicate ratio of the ditance PT and therefore by cor. 1. prop. 45. will make the line of the apides, or, which is the ame thing, the upper apis, to go backward. But at the conjunction and oppoition 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, becaue the force KL is very nearly increaed in the ratio of the ditance PT; decreae in more than the duplicate ratio of the ditance PT; and therefore by cor. 1. prop. 45. caues the line of the apides to go forwards. In the places between the yzygies and the quadratures, the motion of the line of the apides depends upon both thee caues conjunctly, to that it either goes forwards or backwards in proportion to the exces of one of thee caues above the other. Therefore ince the force KL in the yzygies is almot twice as great as the force LM in the quadratures, the exces will be on the ide of the force KL, and by conequence the line of the apides will be carried forwards. The truth of this and the foregoing corollary will be more eaily undertood by conceiving the ytem of the two bodies T and P, to be urrounded on