Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/137

 (by prop. 9. lib. 5. elem.) $$\scriptstyle QT^2$$ and 4SA x QR are equal. Multiply thee equals by $$\textstyle \frac {SP^2}{QR^2}$$, and $$\textstyle \frac {SP^2 \times QT^2}{QR}$$ will become equal to $$\scriptstyle SP^2 \times 4SA$$: and therefore (by cor. 1. and 5. prop. 6.) the centripetal force is reciprocally as $$\scriptstyle SP^2 \times 4SA$$; that is, becaue 4SA is given, reciprocally in the duplicate ratio of the ditance SP. Q. E. I.

From the three lat propoitions it follows, that if any body P goes from the place P with any velocity in the direction of any right line PR, and at the ame time is urged by the action of a centripetal force, that is reciprocally proportional to the quare of the ditance of the places from the centre; the body will move in one of the conic ections, having its focus in the centre of force; and the contrary. For the focus, the point of contact, and the poition of the tangent being given, a conic ection may be decribed, which at that point hall have a given curvature. But the curvature is given from the centripetal force and the bodies velocity given: and two orbits mutually touching one the other, cannot be decribed by the ame centripetal force and the ame velocity.

If the velocity, with which the body goes from its place P, is uch, that in any infinitely mall moment of time the lineola PR may be thereby decribed; and the centripetal force uch as in the ame time to move that body through the pace QR; the body will move in one of the conic ections, whoe principal latus rectum is the quantity $$\textstyle \frac {QT^2}{QR}$$ in its ultimate tate, when the lineolæ PR, QR are