Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/250

174 T; and then, joining CR, there be drawn the right line CP, equal to the abcia CT, making an angle VCP proportional to the ector VCR; and if a centripetal force, reciprocally proportional to the cubes of the ditances of the places from the centre, tends to the centre C; and from the place V there ets out a body with a jut velocity in the direction of a line perpendicular to the right line CV: that body will proceed in a trajectory VPQ. which the point P will always touch; and therefore if the conic ection VRS be an hyperbola, the body will decend to the centre; but if it be an ellipis it will acend perpetually, and go farther and farther off in infinitum. And on the contrary, if a body endued with any velocity goes off from the place V, and according as it begins either to decend obliquely to the centre or acends obliquely from it, the figure VRS be either an hyperbola or an ellipis, the trajectory may be found by increaing or diminihing the angle VCP in a given ratio. And the centripetal force becoming centrifugal, the body will acend obliquely in the trajectory VPQ, which is found by taking the angle VCP proportional to the elliptic ector VRC, and the length CP equal to the length CT, as before. All thee things follow from the foregoing propoition, by the quadrature of a certain curve, the invention of which, as being eay enough. for brevity's I omit.