300 DR. S. CHAPMAN ON THE LAW OF DISTRIBUTION OF MOLECULAR VELOCITIES, 
and that E(U, V, W) must similarly include the term 
(71) 
(c H U 2 + c 22 V 2 + c 33 W 2 + c 23 VW + c a WV + c 12 UV) P (C 2 ), 
where 
(72) 
< 
Cn 
c 
22 
C 33 
= 2 
= 2 
= 2 
0M O 
ov a 
0a: 
0^0 
dw t) 
dz 
dlV 0 
du 0 
dz 
dx 
dw 0 
dz 
C 23 — 2 1 
(dv 0 0w o \ 
\dz ' dy ! 
du 0 
dx ’ 
C 3 i = 3 
(div,, du 0 \ 
\dx dz) 
dv 0 
ay’ 
Cj 2 = 3 i 
/0Wo 
ydy dx)' 
The factor of P (C 2 ) in (71) is equal to 3S' — C 2 S, by (12) and (15), and is therefore 
an invariant with respect to an orthogonal transformation of axes. 
Further since, by (26) and (30), no other derivatives of T and (u 0 , v n , w 0 ) occur in 
AUjCj 24 or AUf'C, 2s , we conclude that none such appear in F (U, V, W)—at any rate, 
to our degree of approximation ; thus the other terms in (12)—(15), while they 
possess the invariant property, do not satisfy the other conditions which must be 
fulfilled by F (U, V, W). 
We therefore conclude that F(U, V, W) is composed only of (70) and (7l) to our 
order of accuracy, and we shall suppose that the two functions P (C 2 ) are expansible 
as power series in C 2 . Throughout this paper we shall assume that all convergency 
conditions necessary for the validity of our analysis are satisfied ; the justification 
of this assumption would offer serious difficulty, and the investigation would lead 
us into regions of pure mathematics which are largely unexplored, and would be 
unsuitable in the present paper. In § 10 we shall see that numerical approximations 
for the most important molecular models confirm the assumption of convergence 
sufficiently for our purpose. 
It is convenient to write our expression for F (U, V, W) in the form 
(73) F(U, V, W) = —B4(u?+Vv+W 
dx 
dry 
0T\ A (2 hm) r 
dz) ^o 1.3.5... (2r + 3)r 
ft-lC* 
(2 hm) r 
—C 0 2hm (c u U 2 + c 22 V 2 +c 33 W 2 +c 2 3 VW + c 31 WU +c 12 UV) 2 ■ \ r , y r C 2r . 
r = o 1 . 3 . 5 ... ( 2 r-f- 5) 
In the first line, when r = 0, the factor r in the denominator is to be omitted. 
The suffix 1 or 2 must be added to m, U, V, W, C, /3, y when we wish to distinguish 
between F 1 (U 1 , V 1} W,) and F 2 (U 2 , V 2 , W 2 ). 
Since, by (72), 
(74) 
C n T C 22 + C 33 — 
