L 5 2 
DR. S. CHAPMAN ON THE KINETIC THEORY OF A COMPOSITE 
and so that the equation for the first approximation to a' 0 becomes (with this value 
of A 0 ) 
(9'03) a .' 0 = — AjAo. 
( c) Second Approximation to a! 0 . 
We may conveniently obtain our second approximation by means of (5"35), taking 
the three rows and columns of V and V' which contain a 00 in the centre. From 
(5‘12)-(5'14) it is easy to see that the elements have the following values (when 
a..o = 1):— 
(9-04) 
(9'05) 
(9*06) 
(9-07) 
^oi^oi — M 2 (^i 1 ) — ^ 10^10 — Pn — Mi (^1 l) — b_i 0 Cfc_ 10 — p. 2 , 
<^1 I a il = I 5Ml“ + n f ( 1 — 2^1 + ~l~k 3 ) + ^ S 5/UiM2^12° + 7TW -~ ^11° (■ = Pi 
V 2 H 2 
'll) 
— — | ifJ -2 + Ml“ ( 1 — 2 ^! + -5-^2) + ^MlM2^12° + -~^22°( = ~P‘22> 
DM 1 
— ^_11®-11 — M1M2 { ^ 2& 1 + 5 ko 2^^12°} — P\2- 
Consequently, the third-order determinant for V is given by 
(9‘08) - 
P 12 pi Pn = -{Pi 2 2 -PnP 22 -^PiP 2 Pi 2 +PnP 2 2 +P 22 Pi 2 } = 
p2 1 Px 
Pi 2 pi Pvi 
and its principal minor by 
( 9 ' 08 l) - p l2 p n = ~{pi 2 -PuPii)- 
I P 22 P 12 
It is easy to prove that 
( 9 ’ 082 ) {Pi2 2 -PnP 2 2) = —2V— (^iD 2 + 2^12^2 + d 2 v 2 2 ), 
V X V 2 
where 
( 909 ) d , EE -i„“{ Jl s 1 (l- 2 /- 1 ^) + -M^A s 0 - 2 e+ 2 i 1 -p J )/« 1 s }. 
M 2 
( 9 '10) d 2 = — h , 2 { J s 2; (1 — 2/x 1( Uv) + MmiM2^i2 U — 2 ( ] 5 + 2&! — 5F) ad 2 } > 
Mi 
(9 ’ 11 ) 2 c £ 12 = t > 4 5 - kiik-A + f (1 — 4 / ci 1 / ct 2 ) (1 — 2 A:, + 5 F,) + 8 mv-kvi ~ 2 ^ +1 F»). 
M1M2 
