Text: 12) The theorem of Fourier referred to in the preceeding article is of such great importance in all questions connected with acoustics that a few words illustrative of it may not be out of place. If a peg is fixed into the rim of a wheel capable of revolving about a fixed center, and at right angles to the plane of the wheel, and if the latter is caused to rotate uniformly and is looked at edgeways the peg will appear to move up and down in a straight line, its velocity being the greatest at the middle of its course, and diminishing as it approaches each end. Under these circumstance the peg appears to perform harmonic vibrations. Now suppose a second wheel, also furnished with a peg in its rim, is made to revolve about the peg of the first as an axis. If the latter is at rest the peg of the second will appear, looked at as above, to perform harmonic vibrations; but if the former is also caused to revolve these vibrations are no longer harmonic, but are the result of adding together the separate harmonic vibrations of the two pegs, in other words of superposing the harmonic vibrations which the second peg performs ifthe first wheel is at rest, upon those which the first peg performs when it is itself in motion. Now it is evident that by continuing this process indefinitely, and by giving the wheels different radii, and different uniform velocities of rotation, the final motion of the last peg looked at sideways as before, would be an exceedingly complicated one, and that an indefinite number of different vibrations could be produced by varying the number, position at starting, radii, and velocities of the wheels, though it could not be assumed without proof that every possible variety could be so produced. This however is what Fourier's theorem asserts, provided that the velocities of rotation of the several wheels of the series are in the proportion of 1, 2, 3, 4, 5, etc. In other words, every periodic vibration is the resultant of a certain number of harmonic vibrations whose periods are one-half, one-third, one-fourth, etc., that of the former.

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Source: 125