SEC. VI11.J OF NATURAL PHILOSOPHY. 169 By the same reasoning, bodies of equal velocities and equal distances from the centre will be equally retarded in their ascent to equal distances. Q.E.D. COR. 1. Therefore if a body either oscillates by hanging to a string, or by any polished and perfectly smooth impediment is forced to move in a curve line ; and another body ascends or descends in a right line, and their velocities be equal at any one equal altitude, their velocities will be also equal at all other equal altitudes. For by the string of the pendulous body, or by the impediment of a vessel perfectly smooth, the same thing will be effected as by the transverse force NT. The body is neither accelerated nor retarded by it, but only is obliged to leave its rectilinear course. COR. 2. Suppose the quantity P to be the greatest distance from the centre to which a body can ascend, whether it be oscillating, or revolving in a trajectory, and so the same projected upwards from any point of a trajectory with the velocity it has in that point. Let the quantity A be the distance of the body from the centre in any other point of the orbit ; and let the centripetal force be always as the power An — ', of the quantity A, the index of which power n — 1 is any number n diminished by unity. Then the velocity in every altitude A will be as v/ P11 — A", and therefore will be given. For by Prop. XXXIX, the velocity of a body ascending and descending in a right line is in tha't very ratio. PROPOSITION XLI. PROBLEM XXVTII. Supposing a centripetal force of any kind, and granting the quadra tures of curvilinear figures, it is required to find as well the trajecto ries in which bodies will move, as the times of their motions in the trajectories found. Let any centripetal force tend to the centre C, and let it be required to find the trajectory VIKAr. Let R, there be given the circle VR, described from the centre C with any interval CV; and from the same centre de scribe any other circles ID, KE cut ting the trajectory in I and K, and the right line CV in D and E. Then draw the right line CNIX cutting the c circles KE, VR in N and X, and the right line CKY meeting the circle VJi 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 anothe body is to fall, so as in the place D to acquire a ve locity equal to the velocity of the first body in I. And things remaining as in Prop. XXXIX, the lineola IK, described in the least given time