112 THE MATHEMATICAL PRINCIPLES [BOOK I, SP2 X PV3 the centripetal force is reciprocally as - — ry^~ — J that is (because AV2 ia given), reciprocally as the square of the distance or altitude SP, and the 3ube of the chord PV conjunctly. Q.E.L The same otherwise. On the tangent PR produced let fall the perpendicular SY ; and (be cause of the similar triangles SYP, VPA), we shall have AV to PV as SP SP X PV SP2 >< PV3 to SY, and therefore -- ^~— - = SY, and - — ^- = SY2 X PV. A V A V And therefore (by Corol. 3 and 5, Prop. VI), the centripetal force is recipSP2 X PV3 rocally as - ~~ry¥~~~ I *na* *s (because AV is given), reciprocally as SP" X PV3. Q.E.I. Con. 1. Hence if the given point S, to which the centripetal force al ways tends, is placed in the circumference of the circle, as at V, the cen tripetal force will be reciprocally as the quadrato-cube (or fifth power) of the altitude SP. COR. 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 re volve in the same circle, and in the same periodic time, about any other centre of force R, as RP2 X 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 RP2 X PT3 to SP2 X PV3; that is, as SP3 X PV3 SP X RP2 to -- p™ — ; or (because of the similar triangles PSG, TPV) to SGS. 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 the same periodic time, about any other centre of force R. as the solid SP X RP2, contained under the distance of the body from the first centre of force S, and the square of its distance from the sec ond 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 fch*3 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.