150 
DR. S. CHAPMAN ON THE KINETIC THEORY OF A COMPOSITE 
We next add X 2 /Xi times the n th row (n > 0) to the (—n) th row; the elements of 
the negative rows (n<;0) thus acquire the following values :— 
(8'07) (m > 0, n<0) 2 mn \a mn + a_ mn +^{a m _ n + a_ m _ n ) [• = 0 
a, 
'00 
^0 
(8'08) (m<0, n<0) S mn (a mn + ^ a m _ n ) = - — mnPl2 (r^). 
These transformations do not alter the value of V (S mn a mn ), nor of V' ( S mn a mn ) when 
applied to this determinant, which continues to be identical with the principal minor 
of V in its new form. Since in each of these determinants the elements of one entire 
quadrant (m > 0, n</0) are all zero, they may be expressed as the product of the 
two simply infinite determinants formed by the quadrants (m > 0, n > 0) and 
(m 0, n <; 0). The former quadrant, however, is the same for V and V', so that 
(5’35) may be written (after a little reduction*) in the form 
(8TJ9) 
“'o = -Vb> 2 (a r -a_ r ) = 
27 
™ 0 ™a,K' 12 (o) V 0 
where V' 0 is the principal minor of V 0 , the general elements a' mn of which is given by 
the equations 
( 8 - 10 ) 
so that 
(8T1) 
a' = S, 
2 7p 12 (m, n) 
mn ^mn 
7rv 1 u 2 2 m+n+!i (m + f ) m+1 (n + f) B+1 K' 13 (0) ’ 
a 00 — 1 m 2 m C t Jc t — a 0m . 
0 
Here, since = m 2 , the equation (4’08) takes the special form 
(g-12) 
v x v 2 
= *£■ 11 C 12 y2 xY 2 [<p k ( y) {B A (r +1, s) + B A (r, s +1)} + 2yty* (y) B A (r, s)] dx dy, 
whence it is clear that V 0 , and consequently also a' 0> is independent of the ratio : v 2 
(i.e., v 1 and v 2 occur only in the form v 1 + p 2 or v 0 ). 
It may readily be verified that a corresponding calculation in the case of V (S mn b mn ) 
leads to the result 
(8-13) /T 0 = 0. 
In the same way we may determine the particular forms assumed by (5‘37), (5‘39), 
and (5’40) in this special case. 
* This reduction chiefly consists in cancelling out factors common to all the elements of ‘corresponding 
rows of Vo and V'o. 
