FREE AND PERFECTLY ELASTIC MOLECULES IN A STATE OF MOTION. 
39 
But a strictly theoretical formula may be deduced if it is permitted to assume that 
the change of temperature between two stations is uniform, whatever that difference 
may amount to, and likewise that the change of density is conformable.* 
Section YI.—On the Velocity with which Impulses are Transmitted 
THROUGH A MEDIUM.t 
§ 39. The reasoning on the subject of this section is founded on the principle that 
the velocity must correspond with the average velocity resolved through the medium 
in any one direction. 
We have seen in §jT7 that the mean square velocity resolved in one direction is 
equal to one-third of that mean square velocity, and it is easy to prove, if all the 
velocities of the molecules are equal, that the average resolved velocity in one direc¬ 
tion is equal to one-half the common velocity. 
As the equal lines representing the molecular velocities on one side of a plane may 
be assumed to radiate equally in every direction from one point, they will spread to 
every point of the hemisphere, resting on the plane; let perpendiculars be dropped 
from these points upon the plane. The quotient of the sum of these divided bj/ their 
number is equal to half the common length of the equal lines. The proof of this is 
derived from the integration of simple circular functions that give the quotient of the 
sum of the sines of a hemisphere divided by their number, or by the surface of the 
hemisphere, equal to half the radius. Thus, if 6 be the inclination of the radiating 
lines to the plane,- v d9 sin 9 cos 0 2-n- represents the aggregate of the perpendiculars 
upon the base of the hemisphere, and d6 cos 9 2-rr represents their aggregate number. 
Collecting the quotients of the first by the second for every value of 9 from 0° to 90°, 
or what is the same, integrating these functions, and dividing the first by the second, 
we have the quotient equal to \v, which is the mean velocity resolved perpendicular to 
the stratum when the molecular velocity v is constant. 
§ 40. But the hypothesis does not admit of the molecules having all the same 
* Note M (barometric formula). 
t [The idea of tbe direct connection between the velocity of sound and that of the molecules is of great 
interest, and leads at once to the conclusion that tbe velocity of sound is independent of density, but 
proportional to absolute temperature. The next person to raise the question was Stefan (‘ Pogg. Ann.’, 
vol. 118, 1863, p. 494), but his calculation is as defective as that of the author. On Waterston’s prin¬ 
ciples, the ratio of the velocity of sound to the molecular velocity of mean square should be \/5/3, as 
was shown by Maxwell (Preston, ‘ Phil. Mag.,’ vol. 3, 1877, p. 453). In the ‘ Philosophical Magazine’ 
for 1858 (vol. 16, p. 481) Waterston returned to the subject. It is curious that he regarded the ordi- 
nary hydrodynamical investigation, not merely as needlessly indirect, but as inconsistent with the 
molecular theory. A result in harmony with experiment cannot be obtained on the basis of a 
hypothetical medium constituted of elastic spheres, for such a medium would have a ratio of specific 
heats different from that observed in gases.—R.] 
