170 
DR. S. CHAPMAN ON THE KINETIC THEORY OF A COMPOSITE 
These are equivalent, so that 
2 
(13-18) ^ = l^r(2 + 
v D 9 tt \ n—l 
r ( 3 — 
2 
2 
= - !+- , 
9-7r\ n—U n—l 
= W + _!_)_* 
9 \ n — l) n—l \ 
2 - 
n—l 
2 
n—l 
2 
1- 
n— 1 
n—l 
2 
n—l 
1 
r i 
n— 1 
n — 1/ • 2x 
sm - 
n- 
by a well-known formula in the theory of gamma functions. When n = 5 (the case 
of Maxwellian molecules), the last equation gives D 0 /D = 1, as it should do. When 
n = oo, corresponding to the case of rigid elastic spheres, D 0 /D = —, as before. 
9x 
It is of interest to consider one or two intermediate values of n in order to see with 
what rapidity our determinantal expression for D 0 /D converges to the value given 
by (13 ’18); we shall not go beyond a third approximation. 
The determinant D, as far as the third row and column, is as follows, where \m 
2 
has been written in place of 
(13-19) D 
1 —m 
n — l 
1 —m 
13 — 4 m + m 2 
5.5 
_ 1 — m 2 
5.7 
23 — 2 7m+ 5m 2 —m 3 
5.5.7 
1—m 2 23 —27m+5m 2 —m 3 433 — 216m + 70m 2 —8m 3 +m' ! 
5.7 
5.5.7 
5.5.7.7 
In the following table are given the first three approximations to D 0 /D for a few 
typical values of n, together with the exact values calculated from (13 ’ 18):— 
Table II. 
n. 
m. 
Do/D. 
Approximations. 
Exact value. 
1st. 
2nd. 
3rd. 
5 
1 
1-000 
1-000 
1-000 
1-000 
9 
h 
1-000 
1-023 
1-027 
1-031 
13 
1 
1-000 
1-039 
1-048 
1-056 
17 
1 
¥ 
1-000 
1-049 
1-060 
1-072 
00 
0 
1-000 
1-083 
1-107 
1-132 
