310 DR. S. CHAPMAN ON THE LAW OF DISTRIBUTION OF MOLECULAR VELOCITIES, 
we have the following equations for A 12 Q X in the two cases under consideration :—- 
(127) - —£ ?—UU — A 1; ,u,c, 2 ' = B i 
1.0.0... yZS I CX )•= 0 
(128) 
S+l 
1 .3.5... (2.S- + 5) 
A 12 U 2 iCr s — C 0 Cn A A 7 ,., {y ri j C 12 ("TiSi) + y r 2 C 12 
r=0 
The corresponding values of A n Qi in the two cases are obtainable from (127), (128) 
by replacing the suffix 2 by 1 throughout. We will write hn (?++) and Cn (ri$i) 
respectively for the values taken by b i2 {riS 1 ) + b 1 2 (r 2 s 1 ) and c 12 (r 1 s 1 ) + c 12 (r 2 5j) when the 
distinction between the suffixes 2 and 1 in these expressions is abolished. In place of 
Hi and iul 2 we now write ^, and /u 12 = fj. 2l = B w 0 = 2m b while B '\ 2 {m, n ) and 
B* 21 (m, n) become identical, and equal to 
(129) “A* (x 2 , if) n A! c (x 2 , if) = B* (m, n ). 
It is convenient to express b n (r x Si) and c n (rj+i) in terms of </>hi (ry) only, eliminating 
<j/\i (ry) and (~//) by means of (115) and a similar equation for <f>" k n (+//). When 
this is done it is found that the coefficient of f n (ry) vanishes for odd values of k, on 
account of the factor ( — l)* in b 12 (r 2 s 1 ) and c 12 (?’ 2 Si). The following are the results 
thus obtained* :— 
(130) Mnsi) = W 
( Tliy ) 
+ 2 if 
k = 1 
J 2 7; + 1 
L4&+1 
[r, s] 
B 2k {r+l,s) + B 2k {v,s+l) 
B” +1 (r,s) + j|U B»-' (r, s)-B“ (r, * 
x 2 f dx dy. 
(131) c i: (r lSl ) = 2 ^ 1 ( T11 y)[B“(r + 2,«) + tB“(r+l,s+l) + B*(r, J> + 2) 
+ Bf 
I 47) +1 
B ak+l {r+l,s) + B 2k+1 (r,s+l 
+ 
2k 
+ 4 y 
47-+1 
f(27- + 2) (27:+ 1) 
1(47: +3) (47)+1) 
(27:+l) 2 
B 2k ~\r +l,s) + B 2k ~\r, s+l))-[ B 2k (r + 1, s) + B 2A (r, s +1)) 
B 2k+2 (r, s) 
+ 
+ 
4T: 2 
(47)+3) (47;+1) (47) +1) (47;-1) 
27; (27:— l) 02 /C -2 / \ 
(47-+l)(47:-l) 1 ' 
_9( ,/ 27 r+_l B2/C+1 ( r , 5 ) + 2^ B2 ,_, ^ 
+ 1 B 2 * (r, «) 
\47;+l 
47;+ 1 
a; 2 // dx dy. 
* In these expressions 4> k u (r n y) is the equivalent, for an encounter between two molecules of the seme 
kind, of <Bi 2 ( r n!/) for molecules of different kinds. Thus (c/. Ill) 
<Bn (rny) = 2 (27: + 1) C R f {1 - P* (cos Xn)}pclp, 
Jo 
where the law connecting x .11 with p and C R may differ from that for yi 2 - Also r n now becomes ( 2 /i?«i) _1 / 2 , 
