OF NATURAL PHILOSOPHY. 171 SEC. VI1LJ COR. 2. So also the angle KIN, in which the trajectory at any place cuts the line 1C. may be readily found by the given altitude 1C of the body : to wit, by making the sine of that angle to radius as KN to IK that is, as Z to the square root of the area ABFD. COR. 3. If to the centre C, and the principal vertex V, there be described a conic section VRS ; and from any point thereof, as R, there be drawn the tangent T RT meeting the axis CV indefinitely pro duced in the point T ; and then joining C CR there be drawn the right line CP, Qequal 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 direc tion 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 VI\ S be an hyberbola, the body will descend to the cen tre ; but if it be an ellipsis, it will ascend perpetually, and go farther and farther off in infinilum. And, on the contrary, if a body endued with any velocity goes off from the place V, and according as it begins either to de scend 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 increas ing 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 ourve, the invention of which, as being easy enough, for brevity's sake I omit. PROPOSITION XLII. PROBLEM XXIX. The law of centripetal force being given, it is required to find the motion of a body setting out from a given place, with a given velocity, in the direction of a given right line. Suppose the same things as in Ihe three preceding propositions; and let the body go off from the place I in the direction of the little line, IK, with the same ve locity as another body, by falling with an uniform centripetal force from the place P, may acquire in I); and let this uniform force be to the force with which the body