SEC. II.] OF NATURAL PHILOSOPHY. 115 QT2 x PC2 ing the extremes and means together, we shall have rfo~ equal to 2BC2 X CA2 — pp — — . Therefore (by Cor. 5, Prop. VI), the centripetal force is 2BC2 X CA2 reciprocally as — — ry~ ; that is (because 2I3C2 X CA2 is given), re ciprocally as-r^v; that is, directly as the distance PC. QEI. I O TJie same otherwise. [n the right line PG on the other side of the point T, take the point u so that Tu may be equal to TV ; then take uV, such as shall be to v G as DC2 to PC2. And because Qr9 is to PvG as DC'2 to PC2 (by the conic sections), we shall have Qv2 •-= Pi' X «V. Add the rectangle n.Pv to both sides, and the square of the chord of the arc PQ, will be equal to the rect angle VPv ; and therefore a circle which touches the conic section in P, and passes through the point Q,, will pass also through the point V. Now let the points P and Q, meet, and the ratio of nV to rG, which is the same with the ratio of DC2 to PC2, will become the ratio of PV to PG, or PV 2DC2 to 2PC : and therefore PY will be equal to „„ — . And therefore the force by which the body P revolves in the ellipsis will be reciprocally as 2 DC2 — ry— X PF2 (by Cor. 3, Prop. VI) ; that is (because 2DC2 X PF2 is I O given) directly as PC. Q.E.I. COR. 1. And therefore the force is as the distance of the body from the centre of the ellipsis ; and, vice versa, if the force is as the distance, the body will move in an ellipsis whose centre coincides with the centre of force, or perhaps in a circle into which the ellipsis may degenerate. COR. 2. And the periodic times of the revolutions made in all ellipses whatsoever about the same centre will be equal. For those times in sim ilar ellipses will be equal (by Corol. 3 and S, Prop. IV) ; but in ellipses that have their greater axis common, they are one to another as the whole areas of the ellipses directly, and the parts of the areas described in the same time inversely: that is, as the lesser axes directly, and the velocities of the bodies in their principal vertices inversely ; :hat is, as those lesser axes dirtily, and the ordinates to the same point %f the common axes in versely ; and therefore (because of the equality of the direct and inverse ratios) in the ratio of equality. SCHOLIUM. If the ellipsis, by having its centre removed to an infinite distance, de generates into a parabola, the body will move in tin's parabola ; and the