Page:Newton's Principia (1846).djvu/118

 the centripetal force is reciprocally as $$\scriptstyle \frac{SP^{2}\times PV^{3}}{AV^{2}}$$; that is (because AV² is given), reciprocally as the square of the distance or altitude SP, and the cube of the chord PV conjunctly. Q.E.I.

On the tangent PR produced let fall the perpendicular SY; and (because of the similar triangles SYP, VPA), we shall have AV to PV as SP to SY, and therefore $$\scriptstyle \frac{SP\times PV}{AV}=SY$$, and $$\scriptstyle \frac{SP^{2}\times PV^{3}}{AV^{2}}=SY^{2}\times PV$$. And therefore (by Corol. 3 and 5, Prop. VI), the centripetal force is reciprocally as $$\scriptstyle \frac{SP^{2}\times PV^{3}}{AV^{2}}$$; that is (because AV is given), reciprocally as $$\scriptstyle SP^{2}\times PV^{3}$$. Q.E.I.

. 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 centripetal force will be reciprocally as the quadrato-cube (or fifth power) of the altitude SP.

. 2. The force by which the body P in the circle APTV 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 $$\scriptstyle RP^{2}\times PT^{3}$$ to $$\scriptstyle SP^{2}\times PV^{3}$$; that is, as $$\scriptstyle SP\times RP^{2}$$ to $$\scriptstyle \frac{SP^{3}\times PV^{3}}{PT^{3}}$$; or (because of the similar triangles PSG, TPV) to SG³.

. 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 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 the 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.