which crest coincides with crest and trough with trough. The joint effect thus produced does not appear in the figure, our object at present being merely to determine the number and positions of the resulting nodes. For the sake of clearness, one set of waves is represented slightly below the other, though in fact the two are strictly coincident.

Let the waves abdf...z be moving from left to right, the waves z'ts'q'...a' from right to left. The crest klm meets the trough pn'm at m. After these have crossed each other, the trough ghk and the crest rq'p will also meet at m, since km and pm are equal distances. Similarly the crest efg and the trough ts'r will meet at m. Accordingly the point m is a node, and, by exactly the same reasoning, so are a, c, e, g, k, p, r, t, &c. The distances between pairs of consecutive nodes are all equal, each being a single pulse-length, i.e. half a wave-length, of either series.

Two pulse-lengths, as gk and km, give three nodes g, k, and m; three pulse-lengths four nodes, and so on. There is thus always one node more than the number of pulses. On the other hand, the fixed ends of the tube, which are the origins of the systems of

reflected waves, occupy two of these nodes. Deducting them we arrive at this result :—

The number of nodes formed is one less than the number of the pulse-lengths (or half wave-lengths), which together make up the length of the vibrating tube.

52. We will now ascertain how the portions of the tube between consecutive nodes move while the

two systems of waves are simultaneously passing along it. Let A and B, Fig. 35, be the fixed ends, as before, and let us take five nodes at the points 1, 2, 3, 4, 5. In (1), the systems of waves coincide, accordingly each point of the tube is displaced through

(

twice as great a distance as if it had been acted on by only one system. The tube thus takes the form indicated by the strong line in the figure. In (2), one set of waves has moved half a pulse-length to the right, and the other the same distance to the left. The two systems are now in complete antagonism, the displacements being equal in amount and opposite in direction at every point. The tube is therefore momentarily in its undisturbed position. In (3), each system has moved through a pulse-length, and the maximum displacements are again produced on the tube, but in opposite directions to those of (1). In (4), where the systems have moved through a pulselength-and-a-half, the tube passes again through its undisturbed position, and, in (5), regains that which it occupied in (1), the systems of waves, meanwhile, having each traversed two pulse-lengths, or one wavelength'. Thus the tube executes one complete vibration in the time occupied by a pulse in passing along a length of the tube equal to twice one of its own ventral segments. In other words, the tube's rate of vibration varies as the number of segments into which it is divided. It moves most slowly in the form shown in Fig. 26 with but a single segment; twice

1

The reader will find that Fig. 35 is rendered more readily intelligible by drawing the two systems of waves in different colours, and the successive positions of the tube in black.

as fast in that of Fig. 27, where it is divided into two segments; three times as fast with three segments, and so on. It is easy to confirm this by direct experiment, the swaying movement of the hand on the tube needing to be twice as rapid for a form of vibration with two segments as for a form with one, and so on.

53.

Instead of comparing the different rates at which the same tube vibrates when divided into different numbers of ventral segments, we may compare the rates of vibration of tubes of different lengths divided into the same number of segments.

Let us take as an example the two tubes AB, CD, Fig. 36, each divided by three nodes into four

ventral segments. By what has been already shown, the time of vibration of either tube will be that which a pulse occupies in traversing two of its ventral segments. Therefore the time of vibration of AB will be to that of CD as A2 is to C2, i.e. as one half of AB is to one half of CD, or as AB is to CD. This reasoning is equally applicable to any other case. Accordingly we have the general result that, when tubes of different lengths are divided into the

same number of ventral segments, their times of vibration are proportional to the lengths of the tubes, or, which comes to the same thing, their rates of vibration are inversely proportional to their lengths. The reader should observe that it has been throughout this discussion assumed that the material, thickness and tension of the tube, or tubes, in question were subject to no variation whatever. Any changes in these would correspondingly affect the rates of vibration produced, but according to less simple laws than variation in length only.

54. We are now prepared to examine the motion of a sounding string. Its ends are fastened to fixed points of attachment and the string is excited at some intermediate point, by plucking it with the finger, as in the harp and guitar, by striking it with a soft hammer, as in the pianoforte, or by stroking it with a resined bow, as in the violin and other instruments of the same class. The impulses thus set up are reflected at the extremities of the string (in the violin at the bridge and at the finger of the performer) and behave towards each other exactly as in the case of the vibrating tube considered above. The results there obtained are, accordingly, at once applicable to the case before us. The string may vibrate in a single segment as in Fig. 26. This is the form of slowest vibration with a string of given