Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/123

The ſame otherwiſe

The perpendicular ST let fall upon the tangent and the chord PV of the circle concentrically cutting the ſpiral are in given ratio's to the height SP; and therefore $\scriptstyle SP^{3}$ is as $\scriptstyle ST^{2}\times PV$, that is (by corol. 3. and 5. prop. 6.) reciprocally as the centripetal force.

Lemma 12.

All parallelograms circmſcribed about any conjugate diameters of a given ellipſis or hyperbola are equal among themſelves.

This is demonſtrated by the writers on the conic ſections.

Proposition X. Problem V.

If a body revolve: in an ellipſis: it is propoſed to find the law of the centripetal force tending to the centre of the ellipſus. Pl. 4. Fig. 1.

Plate 4, Figure 1

Suppoſe CA, CB to be ſemi-axes of the ellippſe; GP, DK conjugate diameters; PF, Qf perpendiculars to thoſe diameters; Qv an ordinate to the diameter GP; and if the parallelogram QvPR be compleated; then (by the properties of the conic ſections) the rectangle PvG will be to $\scriptstyle Qv^{2}$ as $\scriptstyle Qv^{2}$ to $\scriptstyle PC^{2}$ to $\scriptstyle CD^{2}$, and (becauſe of the ſimilar triangles QvT, PCF) $\scriptstyle Qv^{2}$ to $\scriptstyle QT^{2}$ as $\scriptstyle PC^{2}$ to $\scriptstyle PF^{2}$; and by compoſition, the ration of PvG to $\scriptstyle QT^{2}$ is compounded of