Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/113



And, ince the periodic times are in a ratio compounded of the ratio of the radii directly; and the ratio of the velocities inverely; the centripetal forces are in a ratio compounded of the ratio of the radii directly, and the duplicate ratio of the periodic times inverely.

Whence if the periodic times are equal, and the velocities therefore as the radii; the centripetal forces will be alo as the radii; and the contrary.

If the periodic times and the velocities are both in the ubduplicate ratio of the radii; the centripetal forces will be equal among themelves: and the contrary.

If the periodic times are as the radii, and therefore the velocities equal; the centripetal forces will be reciprocally as the radii: and the contrary.

If the periodic times are in the equiplicate ratio of the radii, and therefore the velocities reciprocally in the ubduplicate ratio of the radii; the centripetal forces will be in the duplicate ratio of the radii inverely: and the contrary.

And univerally, if the periodic time is as any power $$\scriptstyle R^n$$ of the radius R, and therefore the velocity reciprocally as the power $$\scriptstyle R^{n - 1}$$ of the radius; the centripetal force will be reciprocally as the power $$\scriptstyle R^{2n - 1}$$ of the radius: and the contrary.

The ame things all hold concerning the times, the velocities, and forces by which bodies decribe the imilar parts of any imilar figures, that have their centres in a imilar poition within thoe figures; as appears by applying the demontration of the preceding caes to thoe. And the application is eay by only ubtituting the equable decription of areas in the place of equable motion and uing the ditances of the bodies from the centres intead of the radii.