Finding the orbits from the focus given

Proposition 46 PROBLEM 32: Any kind of centripetal force, the centre of force and any plane in which the body revolves, being given, and the quadratures of curvilinear figures being allowed.

termine the motion of given place., with a body going off from in, that line a direction the in plane. of given right given velocity

Let S be the centre of SC force, 183 the from the given a body issuing from the place P least distance of that centre P plane, in the direction of the right line PZ, Q, the same body revolving in its trajectory, and PQ,R the trajectory which itself is be found, described in that given plane. Join CQ, Q.S, and if in Q,S we take SV proportional to the centripetal required to force with which the body is attracted to wards the centre S, and draw parallel to CQ, and meeting SC in then will the force SV be resolved into two (by Cor. 2, of the Laws of Motion), the force ST, and the force of which ST aMracting the body in the direction of a line perpendicular to VT T ; TV ; that plane, does not at all change its motion in that But the action plane. f the other force TV, coinciding with the position of the plane itself, at tracts the body directly towards the given point C in that plane c ; t icreftre causes the the force to body move in this plane in the S F were taken away, and same manner ad as if the body were to revolve in free space by means of the force TV alone. But there being given force TV with which the body Q, revolves in free space about the centre C the centripetal about the given centre C, there is given (by Prop. XLII) the trajectory PQ.R which the body describes the place Q,, in which the body will be found at any given time and, lastly, the velocity of the body in that place ; ; Q,. And so e contra. Q..E.I.