796 
PROFESSOR O. REYNOLDS ON CERTAIN DIMENSIONAL 
component velocities as those which enter, them aggregate mass being the same and 
the momentum within the element remaining unaltered, and as the molecules enter 
each successive element in the same manner as they left the preceding, the molecules 
which enter the n th element in a unit of time must have the same mean component 
velocities as those which enter the first; but in the n th element the gas is uniform. 
Therefore, if U, V, W are the component velocities of the uniform gas, when these are 
small so that we may neglect U~, V 3 , V/ 3 
U + u— V + v— 70 + 70 — 
XT_ I Q",r Gx J L 0 XT_ ] - ) cr //(^^) TXT_J . 
2 U + U— 3 2 V + V— 5 2 70 + 7.0— * * * \ / 
0-,1-M — cr.r(M) cr y (jNI) — <r y (M) cr.M — cr-M 
The number of molecules which enter the n th element will also be equal to the 
number which enter the 1 st . 
o o 
— — —~ occ 
Therefore putting ir = ~, and using the dash to indicate the first element 
prx — pa! .(27) 
And the energy carried into the n th is equal to the energy carried into the first 
element. Therefore 
From which equations 
o / /q 
pa — p a 0 . 
a~ — a % and p — p 
(28) 
(29) 
Or the density and pressure of the uniform gas is approximately the same as the 
density and mean pressure of the actual gas. This uniform gas is, therefore, the 
resultant uniform gas according to the definition Art. 78. 
(d) From assumptions IV. and Y. it follows directly that the several changes in 
the inequalities considered separately of each elementary group which enters the 
element in a unit of time will be proportional to the mean inequalities of the group 
as it enters and leaves multiplied by /(a) and by the product of ~ and the mean 
distance through the element traversed by the group. 
Or, as before, putting I for the mean inequality of the group as it enters and leaves 
in respect of G, the separate inequalities are 
-|(G+I) | + I and |(G+I)|+I. 
Whence from assumptions IV. and A", it follows that 
f (G+DS>-=/(«)£M ■ 
(30) 
