358 THE MATHEMATICAL PRINCIPLES [BOOK I.* ridges of the waves is higher than in the unmoved parts of the fluid KL, NO, it will run down from off the tops of those ridges, e, g, i, I, &c., dyfj hj k, &c., this way and that way towards KL and NO ; and because the water is more depressed in the hollows of the waves than in the unmoved parts of the fluid KL, NO, it will run down into those hollows out of those unmoved parts. By the first deflux the ridges of the waves will dilate themselves this way and that way, and be propagated towards KL and NO. And because the motion of the waves from A towards PQ is carried on by a continual deflux from the ridges of the waves into the hollows next to them, and therefore cannot be swifter than in proportion to the celerity of the descent ; and the descent of the water on each side towards KL and NO must be performed with the same velocity ; it follows that the dilatation of the waves on each side towards KL and NO will be propagated with the same velocity ;is the waves themselves go forward with directly from A to PQ,. And therefore the whole space this way and that way towards KL and NO will be filled by the dilated waves rfgr, shis, tklt, v/nnv, &c. Q.E.I). That these things are so, any one may find by making the exper iment in still water. CASE 2. Let us suppose that de, fg, hi, kl, mn, represent pulses suc cessively propagated from the point A through an elastic medium. Con ceive the pulses to be propagated by successive condensations and rarefactions of the medium, so that the densest part of every pulse may occupy a spherical superficies described about the centre A, and that equal intervals intervene between the successive pulses. Let the lines de, fg. hi, Id, &c.. represent the densest parts of the pulses, propagated through the hole BC ; and because the medium is denser there than in the spaces on either side towards KL and NO. it will dilate itself as well towards those spaces KL, NO, on each hand, as towards the rare intervals between the pulses ; and thence the medium, becoming always more rare next the intervals, and more dense next the pulses, will partake of their motion. And because the progressive motion of the pulses arises from the perpetual relaxation of the den?er parts towards the antecedentrnre intervals; and since the pulses will relax themselves on each hand towards the quiescent parts of the medium KL, NO, with very near the same celerity ; therefore the pulses will dilate themselves on all sides into the unmoved parts KL, NO, with almost the same celerity with which they are propagated directly from the centre A; and therefore will fill up the whole space KLON. Q.E.D. And we find the same by experience also in sounds which are heard through a mountain interposed ; and,*if they come into a chamber through the window, dilate themselves into all the parts of the room, and are heard in every corner; and not as reflected from the opposite walls, but directly propagated from the window, as far as our sense can judge. CASE 3 Let us suppose, lastly, that a motion of any kind is propagated