* 
138 
DR. S. CHAPMAN ON THE KINETIC THEORY OF A COMPOSITE 
(5M8) s<-0 
1 B 1 
brs = ~Z » T J 7 TTT7 rT\ { a r + \,s~\~ °b + l,0 + °Coo} 
A 2 A 0 H (r + l;(s+ 1; 
1 Bo 
l ~ ri A 2 A 0 
JR/ (r+l) (s + l) { C * r l ’ S l °b-l, 0 + ^-Oo} 
0 
(5‘19) r>0, s>0 c„ = -C 0 N"„ { P ' n (n$i) +/> , 12 (r 1 Si)} 
Vi 
c rs = - CoN'V u (r 2 Si) r< - 0, s> 0, 
ja 
(5 20) r — 0 , — 0 c rt — — C 0 N rs {/a 22 (r 2 s 2 ) +p 2 f (^ 2 ^ 2 )} 
1'2 
c rs = — C 0 N rs /) 21 (^’ 1 ^ 2 ) — 0. 
1'2 
On the right-hand sides of the above equations the positive numerical values of 
r and s are to be used (whatever their signs on the left-hand side) except when they 
are suffixed to a or 5. 
( b) The Formal Solution of the Equations for a, /3, y. 
If we may solve the linear equations (5'08)-(5*ll), each containing an infinite 
number of unknowns, as if they were finite, we arrive at the results • 
“u — a_o = 
V o («„J 
V (a mn ) 
a r = (r = — co to 7* = co, excluding r = 0), 
V \ a mn) 
S r — Y 7 r f'l >mn } (?■ = — co to r = 00 ? excluding r = 0), 
v (o«») 
_ V r (O 
7r V (c mn ) 
(r = — co to r = + co, including r = ±0). 
In these equations V (a mn ), V ( b mn ), V (c mn ) denote the determinants which have a mn , 
b mn , c mn as their general element; in the two latter, ±m, ±n range from 0 to co, there 
being also a central row and a central column in V ( b mn ) which are not enumerated 
by m or n. In V ( a mn ) the values + 0 of m and n are not distinct from one another, 
so that this also has a central column ( m = 0) and central row ( n = 0); V (c mn ) has 
not got either of these, since ±0 correspond to different rows or columns. The 
determinant V r denotes that obtained from the corresponding V by replacing all the 
elements of the r th column by unity or, in the case only of the central element of the 
7 ,th column of V r (6 mn ), by zero. It may be remarked that all these determinants V 
and V r are infinite in both directions, covering the whole plane. In two quadrants 
(m, 77 . both positive or both negative) the determinants V possess symmetry 
(of (4-23)). 
