left. The two systems are now in complete opposition at every point, and the tube is, therefore, momentarily in its undisturbed position. In (3), each system has moved through a pulse-length, and the combined effect is again produced on the tube, but in the opposite direction to that of (1). In (4), where the systems have moved through a pulse-length and a half, the tube passes again through its undisturbed position, and, in (5), regains the position it occupied in (1), the systems of waves, meanwhile, having each traversed two pulse-lengths, or one wave-length. 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 as fast in that of Fig. 27, when 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

Ar

CH

2

Fig.36

B

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 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.

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 thus obtained are therefore directly 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 length, material and tension. Accordingly, when thus vibrating, the string produces the deepest note of which, all other conditions remaining the same, it is capable. The string may also vibrate in the forms shown in Figs. 27, 28, 35, or in forms with larger numbers of segments. The rapidity of vibration in any one of these forms is, as we have seen [852], proportional to the number of segments formed, so that, with two segments, it vibrates twice, with three, thrice, with four, four times, as fast as in the form with one segment. It follows

hence [43] that the notes obtained by causing a string to vibrate successively in forms of vibration with 1, 2, 3, 4, 5 &c., segments are all partial-tones of one compound sound, the lowest being of course its fundamental-tone.

The modes of eliciting the sounds of stringed instruments described on p. 105 are not capable of setting up any one of the above forms of vibration by itself, but cause several of them to be executed together. The result is that each form of vibration called into existence sings, as it were, its own note, without heeding what is being done by its fellows. Accordingly, a certain number of tones belonging to one family of partial-tones are simultaneously heard.

What precise members of the general series of partial-tones [p. 84] are present, and with what relative intensities, in the sound of a string set vibrating by a blow, depends on the position of the point at which the blow is delivered, on the nature of the striking-object, and on the material of the string. It is clear that a node can never be formed at the point of percussion. Therefore no partialtone requiring for its production a node in that place can exist in the resulting sound. If, for instance, we excite the string exactly at its middle point, the forms of vibration with an even number of ventral segments, all of which have a node at the

centre of the string, are excluded, and only the odd partial-tones, i.e. the 1st, 3rd, 5th, and so on, are heard. In this manner we can always prevent the formation of any assigned partial-tone, by choosing a suitable point of percussion. On the other hand, a vibration-form is in the most favourable position for development when the middle point of one of its ventral segments coincides with the point of percussion. The more nearly it occupies this position the louder will be the corresponding partial-tone, while the more it recedes from this position towards that in which one of its nodes falls on the point of percussion, the weaker will the partial-tone be

come.

The form and material of the hammer, or other object with which the string is struck, have also a great influence in modifying the quality of the sound produced. Sharpness of edge and hardness of substance tend to develope high and powerful overtones, a rounded form and soft elastic substance to strengthen the fundamental-tone. The material of the string itself produces its effect chiefly by limiting the number of partial-tones. The stiffness of the string resists division into very short segments, and this implies, for every string, a fixed limit beyond which further submission becomes impossible; and where, therefore, the series of over