whose lengths' are inversely as the numbers 1, 2, 3, 4, &c. He has further shown that each individual wave-form admits of being thus compounded in only one way, and has provided the means of calculating, in any given case, how many and which members of the series will appear, their relative amplitudes and their differences of phase.

When translated from the language of Mechanics into that of Acoustics, the theorem of Fourier asserts that every regular musical sound is resolvable into a definite number of simple tones whose relative pitch follows the law of the partial-tone series. It thus supplies a theoretical basis for the analysis and synthesis of composite sounds which have been experimentally effected in chapters IV. and V.

When we are listening to a sustained clang, the air at any one point within the orifice of the ear can have only one definite mode of particle-vibration at any one moment. How does the ear behave towards any such given vibration? It proceeds as follows. If the vibration is simple, it leaves it alone. If composite, it analyses it into a series of simple vibrations whose rates are once, twice, three times &c. that of the given vibration, in accordance with Fourier's theorem. In the former event the ear perceives only a simple tone. In the latter, it is able to recognise, 1Arranged in diminishing order.

by suitably directed and assisted efforts, partial-tones corresponding to the rate of each constituent into which it has analysed the composite vibration originally presented to it. The ear being deaf to differences of phase in partial-tones (§ 72), perceives no difference between sounds due to modes of vibration such as those which give rise to the three resultant associated wave-forms shown in Figs. 48, 49, 50, but merely resolves them into the same single pair of partial-tones. Since, however, only one such resolution of a given vibration-mode is possible, the ear can never vary in the group of partial-tones into which it resolves an assigned clang.

The power possessed by the ear of thus singling out the constituent tones of a clang and assigning to them their relative intensities is unlike any corresponding capacity of the eye. Take for instance the two curves shown in Figs. 51 and 52, and try to determine, by the eye alone, what simple waves, present with what amplitudes, must be superposed in order to reproduce those forms. The eye will be found absolutely to break down in the attempt.

We have seen that the loudness of a composite sound depends on extent of vibration, and its pitch on rate of vibration. There remains only one variable element, viz. mode of vibration, to account for the quality of the sound. From this consideration it

follows that some connexion must exist between the quality of a sound and the mode of aerial vibration to which the sound is due. Up to the time of Helmholtz no advance had been made in clearing up the nature of this connexion. It was reserved for him to show that, while no two sounds of different quality can correspond to the same mode of vibration, an indefinitely large number of modes of vibration may give rise to a sound of only one degree of quality. In other words, mode of vibration determines quality, but quality does not determine mode of vibration.

CHAPTER VII.

ON THE INTERFERENCE OF SOUND, AND ON 'BEATS.'

74. In § 71 we examined the principle on which the general problem of the composition of vibrations is solved. We now approach certain very important particular cases of that problem, which it will be worth while to solve both independently and as instances of the method repeatedly applied in § 72.

Suppose that a particle of air is vibrating between the extreme positions A and B while convey

Fig.53.

A

ing a sustained simple tone produced by a tuning-fork, or stopped flue-pipe. Now let a second instrument of the same kind be caused to emit a tone exactly in unison with the first. We will assume that, when the waves of the second tone reach the particle, it is just on the point of starting from A towards B. Two extreme cases are now possible, depending on the movement which the particle would have executed in virtue of the later-impressed vibration alone.

First, suppose

that movement to be from A along the

line AB, either through a greater or less distance than AB, back again to A, and so on. Here the separate effects of the two sets of vibrations will be added together, the particle will, therefore, perform vibrations of larger extent than it would under either component separately. Next, suppose that, under the second set of vibrations alone, the particle would move from A in the opposite direction to its former course, i.e. along BA produced, shown by a dotted line in the figure. In this case the separate effects are absolutely antagonistic; accordingly the joint result is that due to the difference of its components. The particle will, therefore, execute less extensive vibrations than it would have done under the more powerful of the two components acting alone.

The most striking result presents itself when the two systems of vibrations, besides being in opposition to each other, are also exactly equal in extent. In this case the air-particle, being solicited by equal forces in opposite directions, remains at rest, the two systems of vibrations completely neutralising each other's effects. In general, however, these systems, even when equal in extent of vibration, are neither in complete opposition nor in complete accordance, but in an intermediate attitude, so as only partially to counteract, or support, each other. These conclusions