Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/247

 Rh a body acending and decending in a right line is in that very ratio.

Suppoing a centripetal force of any kind, and granting the quadratures of curvilinear figures, it is required to find, as well the trajectories in which bodies will move, as the times of their motions in the trajectories found.

Let any centripetal force tend to the centre C, (Pl. 17. Fig. 4.) and let it be required to find the trajectory VIKk. Let there be given the circle VR, decribed from the centre C with any interval CV and from the ame centre decribe any other circles ID, KE cutting the trajectory in I and K, and the right line CV in D and E. Then draw the right line CNIX cutting the circles KE, VR in N and X, and the right line CKY meeting the circle VR in Y. Let the points I and K be indefinitely near; and let the body go on from V through I and K to k; and let the point A be the place from whence another body is to fall, o as in the place D to acquire a velocity equal to the velocity of the firt body in I. And things remaining as in prop. 39. the lineola IK, decribed in the leat given time will be as the velocity, and therefore as the right line whoe power is the area ABFD, and the triangle ICK proportional to the time will be given, and therefore KN will be ciprocally