Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/130

L x Pv as QR to Pv, that is, as PE or AC to PC; and L x Pv to GvP as L to Gv; and GvP to $\scriptstyle Qv^{2}$ as $\scriptstyle PC^{2}$ to $\scriptstyle CD^{2}$ '; and (by corol. 2. lem. 7.) the points Q and P coinciding, $\scriptstyle Qv^{2}$ is to $\scriptstyle Qx^{2}$ in the ratio of equality; and $\scriptstyle Qx^{2}$ or $\scriptstyle Qc^{2}$ is to $\scriptstyle QT^{2}$ as $\scriptstyle EP^{2}$ to $\scriptstyle PF^{2}$, that is, as $\scriptstyle CA^{2}$ to $\scriptstyle PF^{2}$ or (by lem. 12) as $\scriptstyle CD^{2}$ to $\scriptstyle CB^{2}$. And compounding all thoſe ratio's together, we ſhall have L x QR to $\scriptstyle QT^{2}$ as $\scriptstyle AC\times L\times PC^{2}\times CCD^{2}$ or $\scriptstyle 2CB^{2}\times PC^{2}\times CD^{2}$ to $\scriptstyle PC\times Gv\times CD^{2}\times CV^{2}$, or as 2PC to Gv. But the points Q and P coinciding, 2PC to Gv are equal. And therefore to theſe, will be alſo equal. Let thoſe equals be drawn in $\textstyle {\frac {SP^{2}}{QR^{2}}}$ and $\scriptstyle L\times SP^{2}$ will become equal to $\textstyle {\frac {SP^{2}\times QT^{2}}{QR}}$. And therefore by corol. 1. and 5. prop. 6.) the centripetal force is reciprocally as $\scriptstyle L\times SP^{2}$, that is, reciprocally in the duplicate ratio of the diſtance SP. Q. E. I.

The ſame otherwiſe

Seeing the force tending to the centre of the ellipſis, by which the body P may revolve in that ellipſis. is (by corol. 1. prop. 10.) as the diſtance CP of the body from the centre C of the ellipſis; let CE be drawn parallel to the tangent PR of the ellipſis; and the force, by which the ſame body P may revolve about any other point S of the ellipſis. if CE and PS interſect in E, win be as, $\textstyle {\frac {PR^{2}}{SP^{2}}}$ (by cor. 3. prop, 7.) that is, if the point S is the focus of the ellipſis, and therefore PE be given, as $\scriptstyle SP^{2}$ reciprocally. Q. E. I.