The solution of the problem requires, therefore, the knowledge of do de expressions for the partial differential coefficients and There dz dp are at present no means of finding these by a method entirely a priori; and recourse must consequently be had to experiment and observado dp

tion. To obtain the value of we shall refer to the experimental

determination of the velocity of sound, beginning, first, with some Propositions for finding the velocity theoretically.

PROP. I. To find an expression for the velocity with which a given state of density is propagated in any medium.

The motion is supposed to be in parallel lines. Take an axis parallel to the direction of motion, and let v, p, be the velocity and density of a particle in motion at the distance x from a fixed origin, and at the time t. Then we have the equation,

The differential coefficients are partial with respect to time and space. Let now p' be the density at the same time at the distance x+8x. Then,

After the small time St let the density at the distance x + dx become p. Consequently,

p' − at
= p - de st + &c.

By equating these two values of p' we obtain,

8x

δε

Now is the rate at which the density at the distance x is transferred to the distance x + dx, and is equal to the velocity of the particles + the velocity of propagation.

If the given state of density be propagated with the uniform velocity b, it follows that

an equation applicable to uniform propagation under whatever circumstances it takes place. By integration, v = b. Nap. log p, assuming that v = 0, when p = 1.

Supposing the medium to be such that p = bp, the propagation is known to be uniform and equal to b. Therefore for this medium

the same in any medium which propagates a given state of density with the uniform velocity b, as in the medium defined by the equation p = b2p, provided the motion in both be subject to the condition that = 0 where p = 1.

By help of what precedes the following Propositions may be solved.

PROP. II. To find the impressed accelerative force which will alter the rate of uniform propagation in a medium whose density varies as the pressure.

Let the medium be such that p = a2p, and let b be the altered rate of propagation. It has been shewn that the motion is the same as in a medium for which p = bp, no impressed force acting. Hence the effective accelerative force is the same. Hence

PROP. III. To find the relation between the pressure and the density in a medium which propagates a given state of density uniformly.

It is here supposed that there is no impressed force. The two equations following are therefore to be applied:

Now by supposition a given state of density is propagated with a uniform velocity. Hence if b equal the velocity of propagation, by Proposition I,

dv

= b.

dx

dp pdx'

Integrating,

v = b Nap. log p + $(t);

and introducing the condition that v = 0 wherever p = 1, which can be satisfied when, as we suppose, the propagation is in a single direction only, it follows that (t) = 0. Hence, differentiating with respect to time only,

dv

= b.

dt

dp pdt

Multiplying the first of these by b and subtracting, the result is,

We can now find an expression for the velocity of the propagation of sound in the atmosphere, assuming the velocity to be uniform. Let 9, be the temperature of the air when at rest. Experiments shew that by sudden compression the temperature is increased, and by sudden dilatation diminished. Let 0, + be the temperature corresponding to

any density p at a distance a from the origin, the air being in vibration. Then

The same result may be obtained by means of Proposition II. For we may consider the effect of the heat developed or absorbed by the sudden condensation or rarefaction of the air in vibration to be the same as that of an impressed force, which alters the rate of uniform propagation. The velocity of propagation, supposing the temperature constant and equal to 0,, is a√1 + a0, Hence, by what has been proved,

dp

X = {a2 (1 + a0,) — b2}

pdx

But the effective accelerative force which urges the element pdx

The first term of the right-hand side of this equation is the accelerative force which would act supposing no change of temperature; the other is due to variation of temperature. Consequently,

The vibrations which take place in the propagation of sound are so rapid as not to allow sufficient time for any sensible alteration of the difference of temperature of two contiguous portions of the air by communication of heat from one to the other. This difference may