extremities, whence they are reflected in the opposite direction,

1

t

m

The velocity with which transverse vibrations run along a flexible string is given by the formula y= V, where t denotes tension, m the mass of unit length. The period of a complete vibration is therefore the time required for a pulsation to travel over twice its length: i.e. n =

• wherel is

m

the length of the string, and n the number of vibrations in a second, or

the frequency. Period and frequency are therefore reciprocals.

Fig. 1.-Vibrations of stretched string.

form of vibration is that in which the string vibrates as a whole, and produces its lowest or fundamental note; but it may also be broken up into two or more ventral segments,

separated by nodes or points of rest, the rapidity of vibration being proportional to the number of these segments, and producing partial-tones, which will be described further on.

Nodes.

The breaking-up of the string into a number of nodes with intervening loops or ventral segments may also be determined with certainty by damping the exact point at which one of

the nodes is situated, and exciting the string in the position of a loop or ventral segment. The corresponding nodal points throughout its length will thus be brought to rest, and the

string will sound the upper partial tone belonging to the division. Thus if the damper be applied in the middle, a single nodal point will be formed, the string will vibrate in two halves, and give the octave of its fundamental note. If it be damped at a point one-third from the end, and excited midway between the end and the damping, a second node will be established in the free part; the string will vibrate in three segments, giving the twelfth of its fundamental note.

The higher notes thus obtained are termed harmonics, and will be considered later on. They can easily be shown by throwing a strong light on the string, or by means of small paper riders set astride on it, which are immediately thrown off at the loops, but not at the nodes.

Strings may be excited in various ways, and with corresponding variety in the sound produced. The oldest and simplest mode is obviously that of plucking them, drawing the tense cord out of its position of equilibrium and suddenly letting it go. This is the plan adopted in many ancient and modern instruments, such as the harp and guitar. Or the finger may be armed with a quill or plectrum, as in the case of the zither and in some oriental instruments. In the harpsichord this quill was inserted into a small moveable piece of mechanism termed the jack, which was itself actuated from the keyboard of the instrument.

A third and most important improvement is effected in the pianoforte, where a hammer of comparatively soft material strikes a blow on the string instead of twanging it like the harpsichord jack. But an entirely different course, early adopted, consisted in bringing a bow into frictional contact with the string, and by a succession of impulses conveyed to it, producing a continuous instead of an evanescent tone. In all the innumerable and ancient varieties of the viol and violin family, the bow is made of horsehair rubbed with rosin, kept at a moderate tension by the stick to which it is attached. Many attempts, only partially successful for the most part, have been made to adapt this method of excitation to instruments of the piano species, among which, elastic rollers, and rotating bows of parchment or horsehair passing over rollers, may be named. A current of air directed against a string has long been known to be competent to excite it, and the Aolian harp has been constructed on this principle. Latterly attempts bave been made to render this combination less

The ancient "vielle," now degraded into the hurdy-gurdy, is the oldest, and perhaps the best representative of this class.

vague and fortuitous than it is in that primitive and intractable instrument, but hitherto without producing any very practical result.

It was shown by Delezenne in 1842 that it is impossible to make a string sound if it be excited in the centre by a bow. Duhamel was of opinion that in a string which is giving its foundation tone the first partial is vibrating also, and that since the bow prevents this form of motion, sound cannot be produced. To verify this hypothesis, he endeavoured to sound a string by means of two bows moving in the same direction, to the right and left respectively of the middle point of the string. Still no sound was produced. But on the other hand, if the position of the bows were retained unchanged, and an opposite direction of motion with equal velocity were given them, the foundation tone came out instantly, accompanied by the first upper partial. If the string be attacked successively close to each of the consecutive harmonic points, so as to produce the fundamental tone, the corresponding upper partial is reinforced. At one-third of the length the twelfth has about equal intensity with the fundamental, at one-fourth the double octave, at one-fifth the major seventeenth. The harmonic always slightly precedes the fundamental tone. Speaking generally, a string vibrating transversally can only sound on the condition that it gives two transversal tones, the sharper of which depends on the point of attack, or the mode of excitement.

Sonometer.

The most convenient apparatus for the performance of experiments on strings is the monochord or sonometer. This is a device of great antiquity. It consists, in its most modern form, of a long resonant box bearing on its upper surface wrest-pins, and two bridges set at a fixed distance, usually one or two metres apart. The space between these is occupied by a graduated scale, and a travelling bridge slides along the whole distance. By means of the wrest pins, one or more wires are strained over the fixed bridges to the required tension, and any given length of string can be cut off by the sliding bridge. Besides the wrest-pins, there is usually at one end a pulley grooved to receive the wire, to which weights can be attached so as to verify the second law given above. A modification of this latter contrivance, suited to bear a very considerable load, over two hundredweight in some instances,