7 88 
PROFESSOR 0. REYNOLDS ON CERTAIN DIMENSIONAL 
This fact appeared pointed, but the exact point of it was not at once obvious, nor 
did it fully occur to me until I had completed the investigation founded as already 
described on the assumption. 
Subsequently, however, working at the subject from the other end, so to speak, I 
came to see that whatever might be the action between the molecules, the probable 
effect of encounters in a varying gas would not tend to reduce the molecules after en¬ 
counter to the same state as those of a uniform gas moving with the mean component 
velocities of the varying gas, but to a uniform gas moving with the halves of the 
mean component velocities of all the molecules which cross a unit of surface in a unit 
of time—which pass through an element in a unit of time. 
I had not till then apprehended, nor do I know that it has anywhere been pointed 
out, that the mean component velocities of the molecules which pass through an 
element in a given time are not in the case of a varying gas, as they would be in that of 
a uniform gas (neglecting the squares of the mean component velocities), the doubles 
of the component velocities of the gas. But it turns out to be so (see Art. 77). And 
what is more, these mean component velocities are the very velocities U, V, W, which 
had been found to be necessary as already described. 
The recognition of this fact therefore removed all fundamental difficulty as regarded 
the velocities U, V, W. 
There still, however, remained the question as to whether the molecules might be 
considered to arrive in all respects to the same degree of approximation as from the 
same uniform gas—whether the molecules would arrive in respect to density from the 
same uniform gas as in respect to mean velocity, &c. ; or whether severally in respect of 
density, mean velocity, &c., the uniform gas would correspond to different values of s ? 
The answer to this question depends on the law of action between the molecules, and 
hence it is of necessity left for such light as accrues from the experiments and other 
known properties of gas. 
It is, however, now proved (not altogether from 'first principles, but on certain 
elementary assumptions which might, it is thought, be deduced from first principles) 
that as regards number the molecules will arrive at A as from a uniform gas having 
the density, mean pressure and U, Y, W, of the actual gas at a, certain distance s 
from A, and that as regards mean velocity, mean square of velocity, and mean cube 
of velocity, the molecules will arrive as from uniform gas corresponding in each respect 
with the same or another point. 
So that instead of having one value of s there are four ; the numerical relations 
between which have not been determined from the elementary assumptions, but which 
are all shown to be functions of the temperature and inversely proportional to the 
density, and when the gas varies continuously independent of the direction from 
which the molecules arrive. 
On comparing the theoretical results with those of experiment it is found— 
1 . That the values of s for density and mean square of velocity are equal ; 
