Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/145

in the direction of the line PR; but by virtue of a centripetal force to be immediately turned aſide from that right line into the conic ſection PQ. This the right line PR will therefore touch in P. Suppoſe likewiſe that the right line pr touches the orbit pq in p; and if from S you ſuppoſe let fall on thoſe tangents, the principal latus rectum of the conic ſection (by cor. 1. prop. 16.) will be to the principal latus rectum of that orbit, in a ratio compounded of the duplicate ratio of the perpendiculars and the duplicate ratio of the velocities; and is therefore given. Let this latus rectum be L. The focus S of the conic ſection is alſo given. Let the angle RPH be the complement of the angle RPS to two right; and the line PH, in which the other focus H is placed, is given by poſition. Let fill SK perpendicular on PH, and erect the conjugate ſemi-axe BC; this done, we ſhall have $\scriptstyle SP^{2}-2KPH+PH^{2}$ = $\scriptstyle 4CH^{2}$ = $\scriptstyle 4BH^{2}-4BC^{2}$ = $\scriptstyle {\overline {SP+PH^{2}}}-L\times {\overline {SP+PH}}$ = $\scriptstyle SP^{2}+2SPH+PH^{2}-L\times {\overline {SP+PH}}$. Add on both ſides $\scriptstyle 2KPH-SP^{2}-PH^{2}+L\times {\overline {SP+PH}}$, and we ſhall have $\scriptstyle L\times {\overline {SP+PH}}$ = $\scriptstyle 2SPH+2KPH$, or $\scriptstyle SP+PH$ to PH as 2SP + 2KP to L. Whence PH is given both in length and velocity. That is, if the velocity of the body in P is ſuch that the latus rectum L is leſs than 2SP + 2KP, PH will lie on the ſame ſide of the tangent PR with the line SP; and therefore the figure will be an ellipſis, which from the given foci S, H and the principal axe SP + PH, is given alſo. But if the velocity of the body is ſo great, that the latus rectum L becomes equal to 2PS + 2 KP, the length PH will be infinite; and therefore the figure will be a parabola, which has its axe SH parallel to