142 
DR. S. CHAPMAN ON THE KINETIC THEORY OF A COMPOSITE 
for the elements of the central column in V or V', and defining f_ 0n as equal to — f Qn . 
This means, in effect, that the elements of the central column are to be reversed in 
sign, for the purpose of differencing on the left (in<0). # Now, by definition, 
V' (S 0n a mn ) differs from V (<S 0n a mn ) only in the central row, for which 
f m0 =1 (m > 0), f m0 = -1 (m < 0), and also /_ 00 = -1, 
so that 
<U/mO = 0 if W ^ 0, <bo/o 0 = 1. 
Hence, when transformed into V' ( S mn a mn ), it differs from V ( S mn a mn ) only in the central 
row, all the elements of which are zero except the central one, which is unity. In 
other words, V' (S mn a mn ) is the principal minor of V ($ mn a mn ), and the expression for 
CD 
— 1 (tt r (X_ r ), VIZ., 
o 
(5‘35) 
OD 
2 (a r —a_ r ) = 
0 
v 
/ 
has thus been reduced to a concise symmetrical form. 
(f) A Symmetrical Expression for /3' 0 . 
It appears from (5 ‘21) that 
-v Vo ( ^mn) 
P 0 — A X A 2 _ n v > 
v(6 mn ) 
where V 0 is the same as V, except that all the elements of the central column are 
unity save the central one, which is zero. If we transform V and V 0 by the operations 
described in §5 (d), differencing by columns with m = 1 (for the right) and m = — 1 
(for the left) as starting points, and similarly for the rows, we leave the central 
column (not enumerated by m) untouched as regards the first part, of the operation, 
and the central row untouched by the second part. Thus we obtain the result 
(5-36) /?„ = — XjX, • 
where V (S mn b mn ) is the determinant whose general element is S mn b mn (±rn, ±n ranging 
from 0 to oo), and which has a central column $ 0n b n , a central row S m0 b m , and b as the 
central element. From this determinant we obtain V' 0 (S mn b mn ) by substituting zero 
for* all the elements of the central column except those next to the centre on either 
side (n = ±0) which are replaced by unity. 
* We may note that the elements of the central column are the coefficients of a 0 in (5'01)-(5 - 05), while 
the elements reversed in sign are the coefficients of a_ 0 . 
