Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/138

diminiſhed in infinitum. In theſe corollaries, I conſider the circle as an ellipſis; and I except the caſe, where the body deſcends to the centre in a right line.

Proposition XIV. Theorem VI.

If ſeveral bodies revolve about one common centre, and the centripetal force is reciprocally in the duplicate ratio of the diſtance of places from the centre; I ſay, that the principal latera recta of their orbits are in the duplicate ratio of the area's, which the bodies by radii drawn to the centre deſcribe in the ſame time. Pl. 6. Fig. 1.

Plate 6, Figure 1

For (by cor. 2. prop. 13.) the latus rectum L is equal to the quantity $\textstyle {\frac {QT^{2}}{QR}}$ in its ultimate ſtate when the points P and Q coincide. But the lineola QR in a given time is as the generating centripetal force; that is (by ſuppoſition) reciprocally as $\scriptstyle SP^{2}$. And therefore $\textstyle {\frac {QT^{2}}{QR}}$ is as $\scriptstyle QT^{2}\times SP^{2}$ that is, the latus rectum L is in the duplicate ratio of the area QT x ST. Q. E. D.

Cor. Hence the whole area of the ellipſis, and the rectangle under the axes, which is proportional to it, is in the ratio compounded of the ſubduplicate ratio of the latus rectum, and the ratio of the periodic time. For the whole area is as the area QT x SP deſcribed in a given time, multiplied by the periodic time.