Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/275

 plane, P a body iuing from the place P in the direction of the right line PZ, Q the ame body revolving in its trajectory, and PQR the trajectory it elf which is required to be found, decribed in that given plane. join CQ, QS, and if in QS we take SV proportional to the centripetal force with which the body is attracted towards the centre S, and draw VT parallel to CQ and meeting SC in T; then will the force SV be reolved into two, (by cor. 2. of the laws of motion) the force ST, and the force TV; of which ST attracting 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 of the other force TV coinciding with the poition of the plane it elf, attracts the body directly towards the given point C in that plane; and therefore caues the body to move in this plane in the ame manner as if the force ST were taken away, and the body were to revolve in free pace 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 pace about the given centre C, there is given (by pro. 42.) the trajectory PQR which the body decribes; the place Q in which the body will be found at any given time; and latly, the velocity of the body in that place Q. And o è contra. Q. E. I.