MONATOMIC GAS : DIFFUSION, VISCOSITY, AND THERMAL CONDUCTION. 
103 
all steady states the temperatures are the same, although the gas may be non-uniform 
in temperature, velocity, or composition ; in this case §§ 4-19 give a complete account 
of the first-order phenomena of the gas. 
When the relative proportions of the component gas are varying, so that the 
temperatures are unequal, the determination of the coefficients a, /3, y in §§ 4-9 is 
unaffected, but we must examine how far the physical equations of the succeeding 
sections remain valid. It is immediately evident that all the phenomena grouped 
under the general term “diffusion” (cf. § 10 and §§ 13-16) are uninfluenced by the 
presence or absence of the temperature difference, because the former depend solely 
on the terms in f/(f) 0 which are of odd degree in U, V, W ( cf. § 3 (a)), while the 
terms relating to T' 0 or 
aVo 
dt 
are of even degree. 
Further, it is readily manifest that the equations of viscosity are likewise unmodified. 
The mean hydrostatic pressures of the component gases p u p 2 are indeed altered, but 
their sum p 0 remains constant with the value defined in terms of r 0 and T 0 by (1*18). 
Since the series J (C 2 ) of § 3 (a) is symmetrical in U, V, W, the partial pressures 
(p xz ) i, (p yy )u (Pzz )i are affected equally with p u and similarly for the corresponding 
pressures for the second component gas. The differences p xx —p 0 , Pyy—p<» p zz ~~Pn are 
therefore independent of T' 0 , and it is clear from (3'03), (3'04) that this is true also 
of p yz , p, x , p xy . Hence the equations of viscous stress given in § 11 are applicable 
both to steady and unsteady states of the gas. 
Finally, it may be seen on inspection that the equation of energy, deduced in § 12, 
is true independently of the existence of a temperature difference between the 
component gases. The equations (12'06), (12'07), on which the equation of energy is 
based, remain true in all cases, if T is taken to be T 0 . Also the presence of the even 
power series J (C 2 ) in the expression for /(U, V, W) does not affect the value of UC 2 , 
so that the expressions of § 12 (c) for the coefficient of thermal conduction and the 
specific energy of diffusion are universally true. 
It remains only to form an estimate of the magnitude of T' 0 by determinmg 
approximately the coefficients § r in J (C 2 ). For this purpose it is simplest to make 
use of equation (2‘23), viz., 
( 20 - 01 ) 
(2 hpnf 
1.3.5... (2.S+1) 
AC* = 
ax', 
a t 
o _ 
(2 h 0 m s Y 
, -T AC S * 
1.3.5 ... (2s+l) 
Then it may readily be proved, after the manner of §§ 6, 7 of my second paper ( loc. 
cit.), and as a consequence of the expressions (3'03), (3'04) for /"(l), V, W), that 
( 20 ' 02 ) 
{2h 0 m l ) s 
. ACj 2 * = 32D 0 ^^ 2 s) + d l212 (r, 5 )} S r + d 2U2 (r, s) <L r ], 
1.3.5... (2s + 1) dt r = o 
-- AC 2 2s = 32D 0 ^ 2 [d l221 {r, s) S r + {d 2222 {r, s) + d 2121 {r, s)} <L r ], 
1 .3.5... (2s + 1) dt T = 0 
(20-03) 
