168 
DR. S. CHAPMAN ON THE KINETIC THEORY OF A COMPOSITE 
so that by (7'2l) the general element of D is 
(13-11) 
(r + s + 2) r+s 
(r+f) r (s+f)/ 
r > 0, s > 0. 
The following are the numerical values of the elements as far as the fourth row and 
column, the exact values being given in the first expression, and the decimal values 
to three significant figures in the second :— 
(13-12) D = 
1 
1 
1 
3 
I’OOO 
0*200 
-0-029 
o-oio 
5 
5.7 
5.7.9 
1 
13 
23 
33 
0-200 
0-520 
0'131 
1 
© 
o 
to 
1 — ‘ 
5 
5.5 
5.5.7 
5.5.7.9 
1 
23 
433 
1077 
-0-029 
0-131 
0-353 
0-098 
5.7 
5.5.7 
5.5.7.7 
5.5.7.7.9 
3 
33 
1077 
26613 
O'OIO 
-0-021 
0-098 
0-268 
5.7.9 
5.5.7.9 
5.5.7.7.9 
5.5.7.7.9.9 
By neglecting the rows and columns of D after the first, second, third and fourth 
respectively we obtain the following four successive approximations to D 0 /D :— 
Table I. 
Approximation. 
D. 
D 0 /D. 
- 
1st. 
1-000 
1-0000 
833 
13 
13 
2nd. 
25 =°- 480 
= 1-0833 
235 
4608 
5100 
3rd. 
25.25.49 ~ 0 150 
4608 “ 1 1068 
97 
106168100 
1185408 4 
4th. 
5 6 7 4 92 - 0-035 
1061681 - 1 1165 
The successive approximations to D 0 /D evidently converge with some rapidity 
to the value 1T2 or, more nearly, 1*13; the correction introduced by the second 
approximation covers two-thirds of the error of the first. 
It is of importance and interest to notice that in this case an exact solution is 
obtainable by another method, less general than that of this memoir, but more 
