Finding the orbits from the focus given

PROPOSITION 56 PROBLEM 37

Granting the quadratures of curvilinear figures, and supposing that there are given both the law of centripetal force tending to a given cen tre, and it is the curve superficies whose axis passes through that centre required to find the trajectory which a body will describe in that ; when going off from a given place with in a given direction in that superficies. last construction remaining, let the superficies, and The body T go from the given place a given velocity, S, in the di rection of a line given by position, and turn into the trajectory sought STR, whose ortho BDO is AP. graphic projection in the plane And from the given velocity of the body in the altitude SC, its velocity in any other al titude TC will be also given. moment body describe the particle Tt velocity, in a given With that of time, let the of its trajectory, and let P/? be the projection of that particle described in the plane AOP. Join Op, and a little circle being described upon the curve superficies about the centre with the interval TV be the ellipsis pQ. let AOP the projection of that little circle in the plane because the magnitude of that little circle T/, and And TN or PO CO its distance from the axis is also given, the ellipsis pQ, will be given both in kind and magnitude, as also its position to the right line PO. And since the area PO/? is proportional to the time, and therefore given because the time is given, the angle POp will be given. And thence will be given jo the common intersection of the ellipsis and. the right line Op, together with the angle OPp, in which the projection APp of the tra But from thence (by conferring Prop. XLI, with jectory cuts the line OP. Us 2d Cor.) the mariner of determining the curve APp easily appears. Then from the several points P of that projection erecting to the plane PT AOP, the perpendiculars iven the several points be meeting the curve superficies in T, there will T of the trajectory.