148        AN EXAMINATION  OF THE POSSIBLE EFFECT OF THE
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depression of temperature, is analogous to the increase in the forces of restitution of the particles of air arising from the same cause, to which corresponds an increase in the velocity of propagation of sound.
Another consequence follows from the formula (13), which deserves to be noticed. We have already seen that this formula gives nfjT^1 sec ^ for the value of F, the velocity of propagation. Putting for shortness
l + aj8 = JST...........................(14),
we get from (11) and (12),
.(15).
Kn* + (f + fJ{(K*v? -f (f) (ri^2 + (f)} Hence if q be comparable with n, F, which is a function of the ratio of q to n, will change with n, and therefore the velocity of propagation will depend upon the pitch, which is contrary to observation. But if q be either incomparably greater or incomparably smaller than n, F will assume one or other of its limiting values \/k, VMT; and the velocity of propagation will be independent of the pitch, as observation shows it to be. We are thus led, by considering the velocity of propagation, to the same conclusion as was deduced from the circumstance that sound is capable of travelling to a distance.
Since, then, we are driven to one or other of the alternatives above mentioned, it only remains to decide which we must choose. But before entering on this subject, it will be proper to consider whether the formula (13) is of sufficient generality.
In the first place we may observe, that the formula (13) is only a particular integral of (7). It is adapted to the case in which the motion is kept up by a vibrating plane, which agrees most nearly with the circumstances of ordinary expcrimerits; but a particular law of disturbance as regards the time is assumed, namely, that expressed by a single circular function. Now we know that any periodic function of the time, having r for its period, may be expressed by the sum of a finite or infinite number of circular functions having for their periods r arid its submultiples; and even a non-periodic function may be expressed by a definite integral, of which each element denotes a circular function. So far, therefore, the formula (13) is of sufficient generality.