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I. § 11.J MODES OF VIBRATION. 23
which we mean the numbers of vibrations performed in any given interval of time) are, therefore, proportional to the numbers 1, 2 and 3, which are themselves inversely proportional to the wave-lengths in the three cases respectively. We may express our result thus; The rate of particle-vibration is inversely proportional to the corresponding wave-length. Similar reasoning will apply equally well to any other case; the proposition, therefore, though deduced from the relations of particular waves, holds for waves in general.
The converse proposition admits of easy independent proof as follows. It has been shown (p. 20) that in one period of particle-vibration a wave traverses its own length. This length must therefore, if the velocity of the wave remain constant, be proportional to the period, i.e. inversely proportional to the rate of vibration.
11. We have now connected the extent of the particle-vibration with the amplitude, and its rate with the length, of the corresponding wave. It remains to examine what feature of the vibratory movement corresponds to the third element, the form of the wave.
Suppose that two boys start together to run a