312 DR. S. CHAPMAN ON THE LAW OF DISTRIBUTION OF MOLECULAR VELOCITIES, 
following linear equations connecting the unknown coefficients in the velocity- 
distribution function 
(138) i/3A.= l, 
r = 0 
(139) 2 y,c„ = 1. 
r = 0 
These are true for all values of s from 0 to (», the coefficients b rs and c rs being 
completely determined, in terms of the molecular data, by (130), (l3l), # and (136), 
(137). If we assume that certain convergency conditions are satisfied (138) and (139) 
lead (in the way usual in the case of a finite system of linear equations) to the following 
expressions for ft and y r :— 
(140) ft = V r (b rs )/V (&„), y r = V r (c„)/V (c„), 
where V (b rs ) and V (c„) denote the infinite determinants formed from the arrays (b rs ) 
and (c rs ), thus, 
( 141 ) 
v (b„) s 
K 
^10 
^20 
^30 
<1 
C2 
*4 
III 
Coo 
C 10 
C 20 
C 30 
601 
K 
b'2 1 
^31 
C 01 
c u 
C 21 
C3I 
^02 
2 
b'22 
^32 
C 02 
^12 
C 22 
C 32 
bos 
b\s 
^23 
C 33 
C 03 
C 13 
C23 
C 33 
and V r (b rs ), V r (c rs ) denote the determinants obtained by replacing each element of 
column (r) in V (b rs ) and V (c rs ) respectively by unity. 
The General Expression for the Velocity Distribution Function. 
§ 8 (B) This completes our solution of the fundamental problem of this paper, i.e., 
the determination of the velocity-distribution function for a “ nearly perfect ” simple 
gas, composed of monatomic molecules of the most general type, and which is slightly 
non-uniform as regards temperature and mass-velocity. The solution will be sum¬ 
marized as follows (cf. (10), (73)):— 
(142) /(U,V,W) = 
hm \ s/i 
7r 
-/jm (u 2 +V 2 +W 2 ) 
L y, 1 /.. 8 T .. 3T ...3T\ " (2hm) r 
l B ° T ( U a, v a y W aJ,.?o 1.3.5...( 2 r + 3)r 
- C u (2 hm) (c u U 2 + c 22 V 2 + c 33 W 2 + c 23 V W + c 31 WU + c 12 U V) 
ft-40 2 
2 , 
y 
(2 h?n) r 
, 2,1 
to 1.3.5...(2r+5) yr °7’ 
* The suffix 1 throughout these equations may now be omitted. 
