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

T; and then, joining CR, there be drawn the right line CP, equal to the abſciſſa CT, making an angle VCP proportional to the ſector VCR; and if a centripetal force, reciprocally proportional to the cubes of the diſtances of the places from the centre, tends to the centre C; and from the place V there ſets out a body with a juſt 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 ſection VRS be an hyperbola, the body will deſcend to the centre; but if it be an ellipſis it will aſcend 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 deſcend obliquely to the centre or aſcends obliquely from it, the figure VRS be either an hyperbola or an ellipſis, the trajectory may be found by increaſing or diminiſhing the angle VCP in a given ratio. And the centripetal force becoming centrifugal, the body will aſcend obliquely in the trajectory VPQ, which is found by taking the angle VCP proportional to the elliptic ſector VRC, and the length CP equal to the length CT, as before. All theſe things follow from the foregoing propoſition, by the quadrature of a certain curve, the invention of which, as being eaſy enough. for brevity's ſ I omit.