286 DR. S. CHAPMAN ON THE LAW OF DISTRIBUTION OF MOLECULAR VELOCITIES, 
Except under the differential sign we shall write u = U, v = V, w — W, since we are 
supposing that u 0 = v 0 = w 0 = 0 at the point (x, y, z). The last equation consequently 
reduces to 
( 20 ) 
1 _ _ /0?L ffin 
v dt \0a; 0 y dz J 
In taking mean values of functions of U, V, W, as in (5), we shall neglect the part 
F in the velocity-distribution function f in cases where the mean value is to be 
differentiated or multiplied by a small factor, since the resulting error is only of the 
second order. # Thus, in such cases, we shall write 
( 21 ) 
U 2 C* 
= 1C 2(S+1) , 
U t C 2(s - 1) = iC 2(s+1) , V 2 W 2 C 2 
(s-1) _ 
l_Q2(s + l) 
TT7 
( 22 ) 
C 2s = 1.3.5... (2s+l)(2hm)-, 
while, if either p, q or r is odd, 
(23) UFVWV 7 = 0. 
Since the equation of transfer involves derivatives of the first order only, it is 
sufficient, whenever the mean value of a function of u, v, w is to be differentiated, to 
expand it by Taylor’s theorem in terms of u 0 , v 0 , w 0 , so far as the first degree only; 
if, then, the coefficient of u 0 , v 0 , w 0 is of type (23), the corresponding term may he 
omitted altogether. 
Case I. Q = u (u 2 + v 2 + iv 2 ) s . 
§3 (B) When Q = u(u 2 +v 2 + w 2 ) s , according to the principles just laid down we 
have 
4- {mQ) = v (\j 2 C 2s + 2u u UC 2s + 2sU 2 (m (J U + rqV + w 0 W) C 2(s-1) + ...) 
ox ox 
~ 3 0a> c } 
_ 1.3.5... (2s + 3) 0 ( 1 
0a: \2 hm 
s+l 
. 1 . 3 . 5 ... (2s + 3) / 1 V / _J_ 0 
2 hm) 12 hm dx +1 0a: \2 hm) +Sv dx \2hm)j ’ 
m 
0 , -w. 
0 
(^Q) = 0, — (vwQ) = o, 
* Except in gases of very low density. 
