Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/120

Cor. 1. Hence if the given point S, to which the centripetal force always tends, is placed in the circumference of the circle, as at V; the force will be reciprocally as the quadrato-cube (or fifth power) of the altitude SP.

Plate 3, Figure 4

Cor. 2. The force by which the body P in the circle APTV (Pl. 3. Fig. 4.) revolves about the centre of force S is to the force by which the ſame body P may revolve in the ſame circle and in the ſame periodic time about any other centre of force R, as $\scriptstyle RP^{2}\times SP$ to the cube of the right line SG, which from the firſt centre of force S, is drawn parallel to the diſtance PR of the body from the ſecond centre of force R, meeting the tangent PG of the orbit in G. For by the conſtruction of this propoſition, the former force is to the latter as $\scriptstyle RP^{2}\times PT^{3}SP$ to $\scriptstyle SP^{2}\times PV^{3}$; that is, as $\scriptstyle SP\times RP^{2}$ to $\textstyle {\frac {SP^{3}\times PV^{3}}{PT^{3}}}$ or, (becauſe of the ſimilar triangles PSG, TPV) to $\scriptstyle SG^{3}$.

Cor. 3. The force by which the body P in any orbit revolves about the centre of force S, is to the force by which the ſame body may revolve in the ſame orbit, and in the ſame periodic time about any other centre of force R, as the ſolid $\scriptstyle SP\times RP^{2}$, contained under the diſtance of the body from the firſt centre of force S, and the ſquare of its diſtance from the ſecond centre of force R, to the cube of the right line SG, drawn from the firſt centre of force S, parallel to the diſtance RP of the body from the ſecond centre of force R, meeting the tangent PG of the orbit in G. For the force in this orbit at any point P is the ſame; as in a circle of the ſame curvature