308 DR. S. CHAPMAN ON THE LAW OE DISTRIBUTION OF MOLECULAR VELOCITIES, 
The Integration with respect to C 0 and C K . 
§ 7 (G) In the expressions for h ( rs) and c ( rs ), integrated out with respect to all the 
variables save C 0 and C R , it is now convenient to make the transformation 
(116) 
x 2 = hm 0 C, 2 , if = hm 0 C R 2 . 
In connection with this we shall use the following notation :—■ 
(117) B \ 212 {m, n) = (: 2hm 0 ) m+nm A\f A_\ 2 = (2hmf n+n . ,n A k (p 1 G 2 , 
_ (ZnJrmff, 2p 2 hrn Q G R 2 ) . n A k (2^ 1 /im 0 C 0 2 , 
- “A* (2n\X 2 , 2a,f ). n A! c {2p. x x 2 , 2,ug/), 
• H A /c (miC 0 2 , mAV) 
2 f j. 2 hm a Gf), 
(118) B /c 2 ii 2 (m, n) — (2hm 0 ) m+n . m A k 2l n A k 
12? 
= m A k (2 f i 2 x\ 2,u 1 ?/ 2 ) • "A* (2,U].r 2 , 2//++)- 
We have here used the fact— cf. (96)-(l00)—that n A k (p 2 ,f) is a homogeneous 
polynomial of degree 2 n in p, <x. 
We now use equations (83), (84), (107), (108), in conjunction with §7 (F), to 
write down the following expressions* for h (r, s ), taking particular note of the signs 
of the various terms :— 
(119) h 12 (r lS ,) = 
T +1 * S 4 * 1 
2 [^ u (r^){B*(r+l, s) + W(r, s+1)} 
k = 0 
+ ±fjL 2 y 2 f k 12 ( r 12 y ) B 7 (r, s)] 1212 &V dx dy, 
(120) b 12 (++) = J j 
B k (r+l,s) 
k = 0 
-2 M-VB‘ (r, s) + m+B‘ (r, 5+1)} 
- 4 B*(r, s)] 2112 x’pdxdy. 
In a similar way, from (87), 88), (109), (110) we obtain the following expressions 
for c (r, s) :— 
( 121 ) 
e -<**+y*> 2 [/ 12 ( TvJ y) {B* (r + 2, 5 ) + fB* (r+1, * +1) 
r = o 
+ B 7, (?', s + 2)} + 8 p. 2 f <p r, \ 2 (~ 12 ?/){B 71 (■?• +1, 5) 
+ B k (r, 5+1)} — IOmA/V^h^tiiV) B 7c (r, 5)] 1212 x 2 ?/ 2 rfa; dy. 
* In (119)—(122) the suffixes 1212 or 2112, which should be appended to the symbols B k (in, n )—the 
same for all those within any one square bracket—are for convenience of printing indicated only by 
being placed after the bracket itself. 
