798 
PROFESSOR 0. REYNOLDS ON CERTAIN DIMENSIONAL 
by a gas identical with the resultant uniform gas at a point P, the elementary group 
in the direction BA or PA would arrive at the element with different values as to 
density, mean velocity, &c., from a similar group if the gas were identical with the 
resultant uniform gas at another point in AB. 
Now what the theorem asserts is, that there is some point Pj at which the resultant 
uniform gas is such that the elementary group in direction BA would arrive with 
approximately the same value of N as the actual group, and that the distance P : A is 
independent of the direction of BA, i.e., would be the same for all directions from A. 
In the same way there is some point P 2 at which the resultant uniform gas is such 
that the group of molecules B A would have the same value of v as for the actual group, 
and so for v 2 and v 3 . 
It is not however asserted that AP 1? AP 3 , &c., either are or are not identical. 
Proof of Theorem (17.). 
This follows directly from theorem (I.). 
Taking a series of elements bounded by a cylindrical surface described about the 
element at A and having its axis in the direction of the group, then all the molecules 
of the group leaving one element may be supposed to enter the next. 
In entering the first element at B there will be a difference I between the value of 
G for the actual group and the value of G for the resultant uniform gas. If G B is 
taken for the resultant uniform gas G B + I B will represent the corresponding value for 
the actual gas at B. 
On emerging from the first element G+I for the group will, by theorem (I.), 
8v 
have been diminished by —I, Sr being the thickness of the element, on emerging 
g 
from the next element, G + I will be still further diminished by —I, and so on 
through all the elements, the total diminution of G+I being equal to— 
And by the corollary to theorem (I.), since the variation in the condition of the gas 
is approximately constant, I is approximately constant through the distance s, and 
s will be approximately constant through this distance ; therefore 
f‘-*=I E .(33) 
J 0 S 
Hence, having traversed the distance s, the group will emerge having 
G+I = G B +I B — I B 
= G b . 
(34) 
