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134                    HAND-BOOK OF ACOUSTICS.
at all considerable in proportion to their length, secondary waves are set up, which on reaching the ear give rise to Combination Tones. The higher octave of the fundamental tone, which may be frequently heard, when a tuning-fork is sharply struck, has a similar origin.
Summation Tones.
Helmholtz, as already mentioned, worked out the theory just referred to, mathematically, and proved that two tones with given vibration numbers, may not only produce a third tone, having its vibration number equal to their difference, but also another tone equal to their sum. To this latter sound, the term "Summation Tone " is applied.
It is not difficult to satisfy oneself experimentally of the reality of the summation tones on such an instrument as the Harmonium or American Organ; indeed, these tones are much louder than is generally supposed. Thus if F2 and C| be selected, the summation tone will be A(, which with careful attention may generally be detected; the following table gives in the last column the summation tones that may be produced by the generators in the second column.