wire from its extremity. Helmholtz, who is equally eminent as a mathematician and as an experimental philosopher, has calculated the theoretic intensity of the harmonics developed in various ways; that is to say, the actual vis viva or energy of the vibration, irrespective of its effects upon the ear. A single example given by him will suffice to illustrate this subject. Calling the intensity of the fundamental tone, in each case, 100, that of the second harmonic, when the string was simply pulled aside at a point 4th of its length from its end, and then liberated, was found to be 56.1, or a little better than one-half. When the string was struck with the hammer of a pianoforte, whose contact with the string endured for 3ths of the period of vibration of the fundamental tone, the intensity of the same tone was 9. In this case the second harmonic was nearly quenched. When, however, the duration of contact was diminished to ths of the period of the fundamental, the intensity of the harmonic rose to 357; while, when the string was sharply struck with a very hard hammer, the intensity mounted to 505, or to more than quintuple that of the fundamental tone.*

But why should pianoforte makers strike the middle strings of their instruments at a point from 4th to 4th of their lengths from their extremities? They had no reason for doing so, beyond the fact that the striking of this point gave their ears the greatest satisfaction. The practice, however, is not without a reason. Up to the tones which require these points as nodes, as Helmholtz has pointed out, the overtones all form chords with the fundamental; but the sixth and eighth overtones of the wire do not enter into such chords; they are dissonant tones, and hence the desirability of doing away with them. This is accomplished by making the point at which a node is required

* Lehre von den Tonempfindungen, p. 135.

CURVES DESCRIBED BY VIBRATING PIANO WIRES. 123

that on which the hammer falls. The possibility of the tone forming is thereby shut out, and its injurious effect avoided.

When the door is

The sounds of the Eolian harp are produced by the division of suitably stretched strings into a greater or less number of harmonic parts by a current of air passing over them. The instrument is usually placed in a window between the sash and frame, so as to leave no way open to the entrance of the air except over the strings. Mr. Wheatstone recommends, as a good illustration of this point, the stretching of a first violin string at the bottom of a door which does not closely fit. shut, the current of air entering beneath sets the string in vibration, and when a fire is in the room, the vibrations are so intense that a great variety of sounds are simultaneously produced.* A gentleman in Basel once constructed with iron wires a large instrument which he called the weather-harp or giant-harp, and which, according to its maker, sounded as the weather changed. Its sounds were also said to be evoked by changes of terrestrial magnetism. Chladni pointed out the error of these notions, and reduced the action of the instrument to that of the wind upon its strings.

Finally, with regard to the vibrations of a wire, the experiments of Dr. Young, who was the first to employ optical methods in such experiments, must be mentioned. He allowed a sheet of sunlight to cross a pianoforte wire, and obtained thus a brilliant dot. Striking the wire he caused it to vibrate, the dot described a luminous line like that produced by the whirling of a burning coal in the air, and the form of this line revealed the character of the vibration. It was rendered manifest by these experi

* The action of such a string is substantially the same as that of the syren. The string renders intermittent the current of air. Its action also resembles that of a reed. See Lecture V.

ments that the oscillations of the wire were not confined to a single plane, but that it described in its vibrations curves of greater or less complexity. Superposed upon the vibration of the whole string were partial vibrations, which revealed themselves as loops and sinuosities. Some of the lines observed by Dr. Young are given in

fig. 48. Every one of these figures corresponds to a distinct impression made by the wire upon the surrounding air. The form of the sonorous wave is affected by these superposed vibrations, and thus they influence the clang-tint or quality of the sound.

SUMMARY OF LECTURE III.

The amount of motion communicated by a vibrating string to the air is too small to be perceived as sound, even at a small distance from the string.

Hence, when strings are employed as sources of musical sounds, they must be associated with surfaces of larger area which take up their vibrations, and transfer them to the surrounding air.

Thus the tone of a harp, a piano, a guitar, or a violin, depends mainly upon the sound-board of the instrument.

The following four laws regulate the vibrations of strings:-The rate of vibration is inversely proportional to the length; it is inversely proportional to the diameter; it is directly proportional to the square root of the stretching weight or tension; and it is inversely proportional to the square root of the density of the string.

When strings of different diameters and densities are compared, the law is, that the rate of vibration is inversely proportional to the square root of the weight of the string.

When a stretched rope, or an india-rubber tube filled with sand, with one of its ends attached to a fixed object, receives a jerk at the other end, the protuberance raised upon the tube runs along it as a pulse to its fixed end, and, being there reflected, returns to the hand by which the jerk was imparted.

The time required for the pulse to travel from the hand to the fixed end of the tube and back, is that required by the whole tube to execute a complete vibration.

When a series of pulses are sent in succession along the tube the direct and reflected pulses meet, and, by their⚫

coalescence divide the tube into a series of vibrating parts, called ventral segments, which are separated from each other by points of apparent rest, called nodes.

The number of ventral segments is directly proportional to the rate of vibration at the free end of the tube.

The hand which produces these vibrations may move through less than an inch of space: while by the accumulation of its impulses, the amplitude of the ventral segments may amount to several inches, or even to several feet.

If an india-rubber tube, fixed at both ends, be encircled at its centre by the finger and thumb, when either of its halves is pulled aside and liberated, both halves are thrown into a state of vibration.

If the tube be encircled at a point one-third, one-fourth, or one-fifth of its length from one of its ends, on pulling the shorter segment aside and liberating it, the longer segment divides itself into two, three, or four vibrating parts, separated from each other by nodes.

The number of vibrating segments depends upon the rate of vibration at the point encircled by the finger and thumb.

Here also the amplitude of vibration at the place encircled by the finger and thumb may not be more than a fraction of an inch, while the amplitude of the ventral segments may amount to several inches.

A musical string damped by a feather at a point onehalf, one-third, one-fourth, one-fifth, &c., of its length from one of its ends, and having its shorter segment agitated, divides itself exactly like the india-rubber tube. Its division may be rendered apparent by placing little paper riders across it. Those placed at the ventral segments are thrown off, while those placed at the nodes retain their places.

The notes corresponding to the division of a string into its aliquot parts are called the harmonics of the string.