by neglecting the square, &c. of 002083 on account of its smallness.

Hence

or the velocity increases 0.96 feet for every additional degree of temperature above freezing, and decreases by the same quantity for each degree below freezing.

22. The great difference between the observed velocity of sound and that obtained by theory arises from the developement of heat in the air from the compression which it undergoes.

a. The law of Marriotte, which makes the elastic force of the air proportional to its density, and which has been employed in estimating the elasticity with which each molecule of the aerial column resists condensation, and transmits it to its neighbour, assumes that the temperature of the whole mass of air is alike, and undergoes no change in the act of condensation, and is therefore only true of masses of air which, after compression, are of the same temperature as before. But it is an ascertained fact, that air and all elastic gaseous fluids give out heat in the act of compression, i. e. actually become hotter, a part of their latent heat being developed, and acting to raise their temperature. This is rendered evident in the violent and sudden condensation of air by a tightfitting piston in a cylinder, closed at the end. The cylinder, if of metal, becomes strongly heated; and, if a piece of tinder be enclosed, on withdrawing the piston it is found to have taken fire; thus proving that a heat not merely trifling, but actually that of ignition, has been excited, of at least 1000° of Fahrenheit's scale. Now, when we consider how small

Heat developed in Air in act of Compression.

the mass of air in such an experiment is, compared with that of the including vessel, which rapidly carries off the heat generated, it is evident that if air by any cause could be compressed to the same degree without contact of any other body, a very enormous heat would be generated in it. It would, therefore, resist the pressure much more than if cold; and, consequently, would require a much more powerful force to bring it into that state of condensation than, according to Marriotte's law, would be necessary.

Air, then, when suddenly condensed, and out of contact with conducting bodies, resists pressure more (i. e. requires a greater force to condense it equally) than when slowly condensed, and the developed heat carried off by the contact of massive bodies of its original temperature. In other words, it is under such circumstances more elastic, and our analytical expression for its elasticity must be modified accordingly. The condensation of the aerial molecules in the production of sound is precisely performed under the circumstances most favorable to give this cause its full influence; the condensation being so momentary that there is no time for any heat to escape by radiation; and the condensed air being in contact with nothing but air, differing infinitesimally from its own temperature; so that conduction is out of the question. Let us see now how this will affect the matter in hand.

dx

b. If we denote by the degrees on the scale of Fahrenheit, by which the temperature of the molecules, while in motion and condensed, surpasses that of their quiescent state, the elasticity of the moving molecule is obtained with sufficient accuracy by multiplying E. the elasticity before obtained by 1+r' . 0·002083. dy Now whatever be the law according to which the temperature of a mass of air is increased by a sudden diminution of its volume, it is obvious that for very small condensations, such as those considered in the theory of sound, the rise of temperature will be proportional

to the increase of density. But the increase of density is I

dy

d x'

Effect of the Developement of Heat on the Velocity.

k being a constant coefficient, whose magnitude may become known, either by direct experiment or by the very phenomenon under consideration. The preceding factor becomes, therefore,

1 + '. 0.002083 = 1+k.0.002083 (1 — dy),

x

making

= K

k. 0.002083

dy dx'

K=1+k. 0.002083;

and the elasticity of the condensed molecule is

dx

and the moving force instead of being - Ed. is

dy'

This differs from the expression originally obtained only by the constant factor K. Without, therefore, going again through all the foregoing analysis, we see at once that the general equations of sound will be precisely as before, writing only K. H for H throughout. Making this change then in the expression for the velocity of sound, it becomes

c. The actual numerical value of the constant coefficient K may be determined, as we have before said, in two ways; either by direct experiment on the increase of temperature developed in a given volume of air by a given condensation, or by a comparison of the formula to which we have arrived with the known velocity of sound. As we have already observed, however, the circumstances under which sound is propagated are far more favorable to the free and full production of the whole effect of the cause in

Numerical Value of K.

question than those of any experiments in close vessels. We must not, therefore, be surprised, if the value of K, as derived from such experiments, should differ materially from its value deduced from the velocity of sound; nor vice versâ, if the observed velocity of sound should differ materially from that obtained by calculation, from an experimental value of K. It is sufficient, in a philosophic point of view, to have pointed out a really existing cause, a vera causa, which must act to increase the velocity, and is fully adequate to do so to the extent observed.

d. We have seen that the numerical value of

neglecting K is equal to 916 feet. The observed value on the other hand, is 1090 feet. Hence we have the following equations for determining K and k,

The value of K has also been determined from some ingenious 'experiments made by Messrs. Clement and Desormes to be

K = 1.35,

whence the velocity at the freezing temperature comes out 916 feet 1.35 1064 feet,

which falls short of the actually observed velocity only by about 26 feet. M. Poisson has shown that an absorption of heat in the course of the experiments, which might very well happen, would completely reconcile the observed and theoretical velocities.

Laplace, calculating on the experiments of Messrs. Welter and Gay Lussac, has since obtained a still nearer approximation to the theoretical velocity, the difference amounting only to about 10 feet.

Application of the Formula to the Case of the Gases, &c.

In inquiries of such delicacy, and where the effects of minute errors of experiment become so much magnified, it seems hardly candid to desire a more perfect coincidence.

CHAPTER III.

PROPAGATION OF SOUND IN GASES AND VAPORS.

23. THE analysis, by which we have in the foregoing articles determined the laws and velocity of the propagation of sound in air, applies equally, mutatis mutandis, to its propagation in all permanently elastic fluids, and in vapors, in so far as their properties are the same as those of gases.

The formula (XVII.) expresses, then, the velocity of sound in all such media, provided for (D) we write instead of the density of atmospheric air that of the gas at the freezing temperature, and under the mean pressure h.

In the case of vapors, we must suppose in calculating the value of (D) that they follow the law of gases in their condensation, and that no portion of them undergoes a change of state to a liquid, by reduction to the standard temperature and pressure.

Suppose the specific gravity of atmospheric air to be denoted by s, and that of any gas or vapor under the same temperature by s'; then if and be the velocitics of sound in air and in the gas or vapor we have