370 THE MATHEMATICAL PRINCIPLES [BOOK II. recurs more forcibly ; and therefore is the more increased by the new mo tion impressed at each return. And these are the principal phasr. )mena oi sounds. SECTION IX. Of the circular motion of fluids. HYPOTHESIS. The resistance arising from the want of lubricity in the parts of a fluid, is, casteris paribus, proportional to the velocity with which the parts of the fluid are separated fro?n each other. PROPOSITION LI. THEOREM XXXIX. If a solid cylinder infinitely long, in an uniform and infinite fluid, revolve with an uniform motion about an axis given in position, and the fluid be forced round by only this impulse of the cylinder, and every part of the fluid persevere uniformly in its 'motion ; I say, that the periodic times of the parts of the fluid are as their distances Jrom the axis of the cylinder. Let AFL be a cylinder turning uni formly about the axis S, arid let the concentric circles BGM, CHN, DIO, EKP, &c., divide the fluid into innu merable concentric cylindric solid orbs of the same thickness. Then, because the fluid is homogeneous, the impres sions which the contiguous orbs make upon each other mutually will be (by the Hypothesis) as their translations from each, other, and as the contiguous superficies upon which the impressions are made. If the impression made upon any orb be greater or less on its concave than on its convex side, the stronger impression will prevail, and will either accelerate or retard the motion of the orb, according as it agrees with, or is contrary to, the motion of the same. Therefore, that every orb may persevere uniformly in its motion, the impressions made on both sides must be equal and their directions contrary. Therefore since the impres sions 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 distances of the superficies from the axis. But the differences of