154 
DR. S. CHAPMAN ON THE KINETIC THEORY OF A COMPOSITE 
(d) First Approximation to /3' 0 and 2 — r 2 /3_ r ). 
In this case, owing to the fact that the central element in the numerator 
determinants of (5 *3 6) and (5‘39) is zero, we have to deal with the third order 
determinant in order to obtain a first approximation. From (515)—(518) it is easy 
to see that V (S mn b mn ) may be obtained from V ( S mn a mn ), each being limited to the 
g 
three central rows and columns, if we replace a 00 by -r-^r a 00 , and divide the top and 
A 0 i\ 
bottom rows respectively by Xi and — X 2 . Consequently the third order determinant 
is 
similarly (cf. §5 (f) for the definition of this determinant), 
(9M9) 
obtained from V {S mn b mn ) is equal to Al ' The value ° f li 
Bn 
A 0 R 
P12 
1 
Pll 
1 
( B 0 V 
P12 
n 
Pn 
Ai 
Ai 
AiA 2 r 0 
VA 0 R/ 
Pi 
0 
Pi 
Pi 
0 
Pi 
P22 
a 2 
1 
P \2 
a 2 
P22 
V 2 
P12 
kj — 1 
25A 1 A 2 i'o \-AqR 
JJo 
by (9‘17). Hence, by (5'36), we have 
(9-20) 
of A 0 R k] 1 . A 0 R A^A 2 v / i \* 
P 0 = - -fi -Hr- X — 2r( Vl a r + p 2 a_r)* 
Ro v 0 A l 
v 0 1 
by (918). 
Again, the determinant V 0 (< S mn b' mn ) defined in § 5 ( g) has the form 
(9-21) 
B X2 
JJ 0 
Af.lt/ 
1 Pl»_£s )- P2 
VXi a 2 , ^ 
xIPi 
-F* -1 
Aj Ao 
0 
nf-v 
\ A | A 2 
V\ 
-Pi 
R X2 
A 0 It 
Pi 
*2 
Pi 2 _ P22 
Ai A 2 
1 
0 
Pl_Pi 
A] Ao 
Pi 
Pn _£i8 
Ai A 2 
Bo 
=- 1 (- 4 ^) W (P22-P2 2 ) + ^ 2 {Pl2-PlP2) +^2 2 {Pll-Pl 3 )] 
R X2 
~\a l v 1 2 + 2a' 12 viv 2 + p7, a F 
11 ^' 9 . 9 . 
. 0 
22 
9jVyV 2 \-A-qR 
* This last equation, here proved true only by comparing the first approximations to /3' 0 and 
1r (via. r + >' 2 a -r), may easily be shown to be strictly accurate, by comparing the general expressions for 
these quantities. 
