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 aide from that right line into the conic ection PQ. This the right line PR will therefore touch in P. Suppoe likewie that the right line pr touches the orbit pq in p; and if from S you uppoe let fall on thoe tangents, the principal latus rectum of the conic ection (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 ection is alo 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 poition. Let fill SK perpendicular on PH, and erect the conjugate emi-axe BC; this done, we hall have $$\scriptstyle SP^2 - 2KPH + PH^2$$ = $$\scriptstyle 4CH^2$$ = $$\scriptstyle 4BH^2 - 4 BC^2$$ = $$\scriptstyle \overline {SP + PH^2} - L \times \overline {SP + PH}$$ = $$\scriptstyle SP^2 + 2SPH + PH^2 - L \times \overline {SP + PH}$$. Add on both ides $$\scriptstyle 2KPH - SP^2 - PH^2 + L \times \overline {SP + PH}$$, and we hall 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 uch that the latus rectum L is les than 2SP + 2KP, PH will lie on the ame ide of the tangent PR with the line SP; and therefore the figure will be an ellipis, which from the given foci S, H and the principal axe SP + PH, is given alo. But if the velocity of the body is o 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