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



And by a like reaoning, a body will be moved in an ellipis, 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.

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

Suppoe the indefinitely mall angle PSQ to be given; becaue then all the angles are given, the figure SPRQT will be given in pecie. Therefore the ratio $$\textstyle \frac {QT}{QR}$$ is alo given, and $$\textstyle \frac {QT^2}{QR}$$ is as QT that is (becaue the figure is given in pecie) as SP. But if the angle PSQ is any way changed, the right line QR, ubtending 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 ame 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 ditance SP. Q. E. I.