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



The ame otherwie

The perpendicular ST let fall upon the tangent and the chord PV of the circle concentrically cutting the piral 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.

All parallelograms circmcribed about any conjugate diameters of a given ellipis or hyperbola are equal among themelves.

This is demontrated by the writers on the conic ections.

If a body revolve: in an ellipis: it is propoed to find the law of the centripetal force tending to the centre of the ellipus. Pl. 4. Fig. 1.



Suppoe CA, CB to be emi-axes of the ellippe; GP, DK conjugate diameters; PF, Qf perpendiculars to thoe diameters; Qv an ordinate to the diameter GP; and if the parallelogram QvPR be compleated; then (by the properties of the conic ections) the rectangle PvG will be to $$\scriptstyle Qv^2$$ as $$\scriptstyle Qv^2$$ to $$\scriptstyle PC^2$$ to $$\scriptstyle CD^2$$, and (becaue of the imilar triangles QvT, PCF) $$\scriptstyle Qv^2$$ to $$\scriptstyle QT^2$$ as $$\scriptstyle PC^2$$ to $$\scriptstyle PF^2$$; and by compoition, the ration of PvG to $$\scriptstyle QT^2$$ is compounded of