160 
DR. S. CHAPMAN ON THE KINETIC THEORY OF A COMPOSITE 
§ 12 . The Equation of Energy. 
(a) The Deduction of the Equation. 
In order to obtain the equation of energy we return to the consideration of the 
equation of transfer (2'0l). Having determined the form of the velocity distribution 
function/(U, V, W) correct to the first order of small quantities, we can now make 
the equations of transfer accurate so far as the second order, and this is necessary in 
connection with the equation of energy. Also we need no longer suppose that the 
mean motion of the gas (u 0 , v 0 , w 0 ) at the point considered is zero. 
We add to (2'0l) the corresponding equation for the molecules of the second kind, 
with the result 
( 12 - 01 ) 
g^(DQi + r 2 Q 2 ) + 
{ v i ( w )iQi ~t (^aQs} 
_ U v / ^Qi \ 
W 1 \d(u)J 
In this equation we substitute in turn Q = 1, Q =m (u), and Q = \m {(u) 2 + (r) 3 + (w) 2 }, 
thus obtaining the equations of density, momentum, and energy. Since all three of 
these quantities are conserved unchanged during the molecular encounters, we have 
in each case A (Q : + Q 2 ) = 0. The first equation takes the form 
( 12 - 02 ) 
Srp . d VqVq dv ^W^ _ 
dt dx dy dz 
where we have made use of the notation of § 1. 
We will denote the operator 
a ^ a ^ .a ^ a 
- + w 0 — + v 0 — +W 0 — 
dt dx dy dz 
D 
by —, commonly known as the ‘ ‘ mobile operator ” ; if q is any function of 
xJ 1/ 
x, y z, t> ^ denotes the rate of change, with respect to time, if the value of q at 
a point which moves with the mean velocity of the gas. Then the equation of 
continuity (12’02) may also be written in the form 
(12-03) 
1 Dj/ 0 fdu, dv Q dlV, 
Vo 
Dt 
+ 
+ 
+ 
dx dy dz 
= 0. 
The second equation, the equation of momentum, may be written as 
(12-04) ^ t (p 1 u 1 + p 2 u 2 ) + — {p 1 {u)i 1 + p 2 (u) 2 2 } + ^ {pi (u\ {vJ + p.j (u) 2 (v 2 )} 
+ 5-{pi(^)i( w )i+P2(w) 2 (w)J -(riXi + ^Xa) = 0, 
dz 
