SEC. X.J OF NATURAL PHILOSOPHY- 183 Let S be the centre of force, SC the least distance of that centre from the given plane, P a body issuing from the place P in the direction of the right line PZ, Q, the same body revolving in its trajectory, and PQ,R the trajectory itself which is required to be found, described in that given plane. Join CQ, Q.S, and if in Q,S we take SV proportional to the centripetal force with which the body is attracted to wards the centre S, and draw VT parallel to CQ, and meeting SC in T ; then will the force SV be resolved into two (by Cor. 2, of the Laws of Motion), the force ST, and the force TV ; of which ST aMracting the body in the direction of a line perpendicular to that plane, does not at all change its motion in that plane. But the action c 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 ; a«d t icreftre causes the body to move in this plane in the same manner as if the force S F were taken away, and the body were to revolve in free space about the centre C by means of the force TV alone. But there being given the centripetal force TV with which the body Q, revolves in free space 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. PROPOSITION XLV1L THEOREM XV. Supposing the centripetal force to be proportional to t/ie distance of the body from the centre ; all bodies revolving in any planes whatsoever will describe ellipses, and complete their revolutions in equal times ; and those which move in right lines, running backwards and forwards alternately ', will complete ttieir several periods of going and returning in the same times. For letting all things stand as in the foregoing Proposition, the force SV, with which the body Q, revolving in any plane PQ,R is attracted to wards the centre S, is as the distance SO. ; and therefore because SV and SQ,, TV and CQ, are proportional, the force TV with which the body is attracted towards the given point C in the plane of the orbit is as the dis tance CQ,. Therefore the forces with which bodies found in the plane PQ,R are attracted towaitis the point O, are in proportion to the distances equal to the forces with which the same bodies are attract-ed every way to wards the centre S ; and therefore the bodies will move in the same times, and in the same figures, in any plane PQR about the point C. n* they