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 theſe 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, becauſe 4SA is given, reciprocally in the duplicate ratio of the diſtance SP. Q. E. I.

Cor. 1. From the three laſt propoſitions 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 ſame time is urged by the action of a centripetal force, that is reciprocally proportional to the ſquare of the diſtance of the places from the centre; the body will move in one of the conic ſections, having its focus in the centre of force; and the contrary. For the focus, the point of contact, and the poſition of the tangent being given, a conic ſection may be deſcribed, which at that point ſhall 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 deſcribed by the ſame centripetal force and the ſame velocity.

Cor. 2. If the velocity, with which the body goes from its place P, is ſuch, that in any infinitely ſmall moment of time the lineola PR may be thereby deſcribed; and the centripetal force ſuch as in the ſame time to move that body through the ſpace QR; the body will move in one of the conic ſections, whoſe principal latus rectum is the quantity $\textstyle {\frac {QT^{2}}{QR}}$ in its ultimate ſtate, when the lineolæ PR, QR are