148 
DR. S. CHAPMAN ON THE KINETIC THEORY OF A COMPOSITE 
equal to the product of the two remaining quadrants ; in the positive quadrant the 
determinants are the same (r > 0 , s> 0 ). Consequently in this case (5‘35) becomes 
(7‘20) 
00 
a' 0 — — XjXs 2 (a r — a_ r ) = — 
27 
_D 0 
2 ™ 0 m 2 A 0 K 12 (h) D 
where D is the determinant whose general element is 
(7-21) 
5 n _ ? (r + s + f) r+ , 7 
" " ~ " (*■ + *),(*+«. 
+ S> 
r > 0 , s > 0 , 
and D 0 is its principal minor. 
Similarly, it follows from (5'36) and (7'16)-(7‘19) that /3' 0 can be likewise expressed 
as the quotient by an infinite determinant (covering only a quadrant of the infinite 
plane) of its second minor, as follows :— 
(7-22) 
£'o= - 
27R,A 2 
Dh 
27n/ 0 m 2 B 0 K' 12 (O)D' ' 
Here D' is the determinant whose general element d rs is given by 
(7'23) (r > 0, s > 0) d„ = Kt-K-h + l- 
■ — X ( r + lj? + f)r + » 1, 
‘r* "(r + f) r (* +f)/ r+M 
(7-24) r >0 d r0 = d 0r = A-i - (K- 1) = - KK d 00 = 1. 
r r 
Thus it is the same as the determinant D of (7'20) except that the r th row and 
column (r !> 1 ) are divided throughout by r. The determinant Dh is the second 
minor of D', i.e., the minor of the next element to the centre in the first row or 
column. 
Again, from (5’37), we may prove that 
(7'25) 
CO 
2 r (p 1 a r + v 2 a._ r ) = 
1 
27r 0 ~D X 
27n/ 1 7n 2 A 0 K / 12 (0) D 
where Di is the second minor of D just as D'j is of D'. 
It is easy to see that* 
(7 - 26) 
H! _ Dh 
D D' ' 
Likewise, from (5'39), we have 
(7'27) 
2 {vifl r + v 2 ft_ r ) 
1 
271 0 dt 
27rv l m 2 B 0 K\ 2 (0) D" ’ 
* Of. the footnote on p. 154, indicating that u 2 cc_ r ) is, quite generally, a mere multiple of yS' 0 . 
