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

 diminihed in infinitum. In thee corollaries, I conider the circle as an ellipis; and I except the cae, where the body decends to the centre in a right line.

If everal bodies revolve about one common centre, and the centripetal force is reciprocally in the duplicate ratio of the ditance of places from the centre; I ay, 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 decribe in the ame time. Pl. 6. Fig. 1.

For (by cor. 2. prop. 13.) the latus rectum L is equal to the quantity $$\textstyle \frac {QT^2}{QR}$$ in its ultimate tate when the points P and Q coincide. But the lineola QR in a given time is as the generating centripetal force; that is (by uppoition) 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.

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