372 THE MATHEMATICAL PRINCIPLES [BOOK 11. farther, its impulse will carry the outmost cylinder also about with it, Tinless the cylinder be violently detained; and accelerate its motion till the periodic times of both cylinders become equal among themselves. But if the outward cylinder be violently detained, it will make an effort to retard the motion of the fluid ; and unless the inward cylinder preserve that mo tion by means of some external force impressed thereon, it will make it 3ease by degrees. All these things will be found true by making the experiment in deep standing water. PROPOSITION LIL THEOREM XL. If a solid sphere, in an uniform and infinite fluid, revolves about an axis given in position with an uniform motion., and thejiuid be forced round by only this impulse of the sphere ; and every part of the fluid perse veres uniformly in its motion ; I say, that the periodic times of the parts of the fluid are as the squares of their distances from the centre of the sphere. CASE 1. Let AFL be a sphere turn ing uniformly about the axis S, and let the concentric circles BGM, CHN, DIO, EKP, &cv divide the fluid into innu merable concentric orbs of the same thickness. Suppose those orbs to be solid ; and, because the fluid is homo geneous, the impressions which the con tiguous orbs make one upon another will be (by the supposition) as their translations from one another, and the contiguous superficies upon which the impressions are made. If the impression upon any orb be greater or less upon its concave than upon its convex side, the more forcible impression will prevail, and will either accelerate or retard the velocity of the orb, ac cording as it is directed with a conspiring or contrary motion to that of the orb. Therefore that every orb may persevere uniformly in its motion, it is necessary that the impressions made upon both sides of the orb should be equal, and have contrary directions. Therefore since the impressions are as the contiguous superficies, and as their translations from one another^ the translations will be inversely as the superficies, that is, inversely as the squares of the distances of the superficies from the centre. But the differ ences of the angular motions about the axis are as those translations applied to the distances, or as the translations directly and the distances inversely; that is, by compounding those ratios, as the cubes of the distances inversely. Therefore if upon the several parts of the infinite right line SABCDEQ