156 
DR. S. CHAPMAN ON THE KINETIC THEORY OF A COMPOSITE 
When the molecules are regarded as rigid elastic spheres of radii <r, and <r 2 , it may 
readily be shown (cf. ‘ Phil. Trans.,’ A, vol. 216, § 9, p. 320) that 
(9'31) 
(9-32) h, 
k'»(d)= 
_ (^ + 2 ) t j, t __ 2d, V i j /, t _ / 2<t 2 \ 2 7 t j. t _ 5 (^ + 3) <+1 
— -rr "'ll — "U2 K 22 — / "U2 "U2 — 3 / . C \ 
{t + f )t 
<Ti + cr 2 . 
-O'! + cr 2 . 
+ f) 
i + 1 
th 
When the molecules are point centres of force proportional to the inverse n 
power of the distance, the force at unit distance being K 12 mj??!- 2 , it may be shown* 
that 
hm 1 rn 2 V ~ 1 L 
m 1 + m 2 ) 
(9-33) 
where 
(9-34) 
and 
(9’35) 
K' 1 ,(0) = 4I,(«) 
7r 
[K 12 (m 1 +m 2 )y-- 1 r 3 
1 
71-1/’ 
f- 
Ii (n) EE 4?r sin 2 . a 
Jo 
da 
-0 = 2 
dt) 
> 7 ft being the root of the following equation in : 
n-l | ’ 
} 
(9-36) 
Further, 
(9-37) 
where 
(9-38) 
Also 
(9-39) 
W--A(2-l =0- 
n— 1 \a 
n— 1 
L = 
ru+3--^- 
77—1 
(^+f) t r (3 — 
n— 1 
^ 5 - 
r U+4- 
+l)t +1 r (3 
77—1/ 1 ,( 77 ) 
" Ii M ’ 
77—1 
I 2 ( 77 ) = 7r sin 2 0 . a (7a. 
Jo 
k (_ /k„v- , t 
"'ll — 
K a2 
7’ * 
' v 12 
u = ilA" fc u ‘, 
K,, 
where K n , K 22 are the force constants appropriate to a pair of molecules of the first 
or second kind respectively; we here assume that the law of variation of force with 
distance is the same whether the molecules are like or unlike—if this be not so, n must 
* Cf. ‘ Phil. Trans.,’ A, vol. 216, § 9, p. 320, or Jeans’ 1 Dynamical Theory of Gases’ (2nd edit.), §§ 305 
et seq. I have adopted the notation of the above equations in order to facilitate comparison with the 
corresponding equations in Jeans’ treatise, 
