322 DR. S. CHAPMAN ON THE LAW OF DISTRIBUTION OF MOLECULAR VELOCITIES 
and 
71 — 5 
(186) K„ = 2tt-\A 1 ({(« +f) 1+ ,}r (« + 4 ■ 
n— 1 
t + 3 — 
n— 1 
(^+f )t 
K 0 ,i. 
Hence the values assumed by B,, and C 0 in this special case are as follows (cf. 
(170), § 8(D)):— 
(187) 
B a = 
(2 km)' 1 
71—5 
71 — 1 
15 3 
32 AA,r 4— 
n— 1 
C 0 = 
(2 Am) 1 ^ 
71—0 
1 
375 
8 (4 — 1 v 
From (186) we have 
A — c r0 — / c r /«- 0 — 2 r 2 r G t K til / K 0il 
t = 0 
r t 
t -f- 3 — 
= 2~ r 2 — 
t=oti (t+f) 
= 2 “ r F f-r, 4 
n— 1 
n — 1 
? 2 "? 
■i > 
in the notation of the hypergeometric function. It may hence be shown, without 
much difficulty, that (if n > 5) b r0 and c r0 steadily increase to infinity with r, the 
n —5 
rate of increase being comparable with that of r n ~ x . 
Since the functions </> 2k ( Tiy) all depend on h in the same way, it is clear that, with 
the above values of B 0 and C 0 , the elements h rs and c rs and consequently, also, the 
coefficients (3 r and y r in the velocity-distribution function for molecules of this type, 
are independent of h, i.e., they are independent of temperature. They are, indeed, 
pure numbers, depending only on the molecular mass and on the force constant of the 
molecules. 
It is of interest to determine the value of the elements S r0 b r0 (or S r0 c r0 ) of the outer 
row or column of V ( S rs b rs ), in this special case. We have, by (175), 
GAo = GoGo = (-2)“ r 2 (-l) i r C t K t ,i/K 0 ,i, 
t = 0 
t + 3- —— 
= ( — 2)~ r 2 (— iTT \_ n^_la 
1 ^ 1 (t+ f) t 
t = 0 
= (_2)-F(_ r ,4- — 1), 
in the notation of hypergeometric functions, or, in terms of gamma-functions, 
(188) SJ>„ = J rt c,„=(-2)- 
r(}) r r-i+ 
71— 1 
r(r+j)r 
n— 1 
= (- 2 )“ 
r—1 + 
n— 1 
(»• + 4), 
