Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/127

 $$\scriptstyle \frac {2DC^2}{PX} \times PF^2$$ (by cor. 3. prop. 6.) that is, (becaue $$\scriptstyle 2DC^2 \times PF^2$$ is given) directly as PC. Q. E. I.

And therefore the force is as the diŧance of the body from the centre of the ellipis; and vice versa if the force is as the ditance, the body will move in an ellipis whoe centre coincides with the centre of force, or perhaps in a circle into which the ellipis may degenerate.

And the periodic times of the revolutions made in all ellipes whatoever about the ame centre will be equal. For thoe times in imilar ellipes will be equal (by corol. 3 and 8. prop. 4.) but in ellipes that have their greater axe common, they are one to another as the whole areas of the ellipes directly, and the parts of the areas decribed in the ame time inverly; that is, as the leer axes directly. and the velocities of the bodies in their principal vertices inverely; that is. as thoe leer axes directly, and the ordinates to the ame point of the common axis inverely; and therefore (becaue of the equality of the direct and invere ratio's) in the ratio of equality.

If the ellipis by having its centre removed to an infinite ditance degenerates into a parabola, the body will move in this parabola; and the force, now tending to a centre infinitely remote, will become equable. which is Ga1ileo's theorem. And if the parabolic ection of the cone (by changing the inclination of the cutting plane to the cone) degenerates into an hyperbola, the body will move in the perimeter of this hyperbola, having its centripetal force changed into a centrifugal force. And in like manner as in the