Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/122

Scholium.

And by a like reaſoning, a body will be moved in an ellipſis, or even in an hyperbole, or parabola, by a centriperal force which is reciprocally as the cube of the ordinate directed to an infinitely remote centre of force.

Proposition IX. Problem IV.

If a body revolves in a final PS, cutting all the radii SP, SQ, &c. in a given angle: it is propoſed to fund the law of the centripetal force tending to the centre of that ſspiral. Pl. 3. Fig. 6.

Plate 3, Figure 6

Suppoſe the indefinitely ſmall angle PSQ to be given; becauſe then all the angles are given, the figure SPRQT will be given in ſpecie. Therefore the ratio $\textstyle {\frac {QT}{QR}}$ is alſo given, and $\textstyle {\frac {QT^{2}}{QR}}$ is as QT that is (becauſe the figure is given in ſpecie) as SP. But if the angle PSQ is any way changed, the right line QR, ſubtending the angle of contact QPR, (by lem. 11.) will be changed in the duplicate ratio of PR or QT. Therefore the ratio $\textstyle {\frac {QT^{2}}{QR}}$ remains the ſame as before, that is a SP. And $\textstyle {\frac {QT^{2}\times SP^{2}}{QR}}$ is as $\scriptstyle SP^{3}$, and therefore (by corol. 1. and 5. prop. 6.) the centripetal force is reciprocally as the cube diſtance SP. Q. E. I.