132 
DR. S. CHAPMAN ON THE KINETIC THEORY OF A COMPOSITE 
as r -1 N„. The meaning of the symbols (x + -f) r+1 , (s + §) 4+1 will be understood from 
the following definition, where q is a positive integer and p any number whatsoever : 
(4’06) 
p q =p(p-i)(i>-2 ) ... (p-g + i), 4>o = l. 
We shall have frequent occasion to use this factorial symbol. 
The following are the expressions found for p ( rs ) and p ( rs) in the various 
cases :— 
(4'07) p n (nsi) = || e {x ' +vl) xY 
£{r+l, s + l) 
2 <pn k (y) B 2A (r+l, s) + B“(r, s + l) 
k = 1 
+ 2 y' 
\ i 2k+1 B2 , + i( r} s )+ s)-B 2 * (r, s 
l4fc+l 
4& + 1 
dx dy, 
(4'08) pu{ri8i) = ^^2 || e {x '+^x 2 y 2 
(r+l, s + l) 
2 [0i2* (y) {B A (r +1, s) + B*(r, s +1)} 
k = 0 
+ 4/* 2 3/fyi a %) P/ (r, s)] 1212 dx dy. 
(4-09) Pl2 (r 2 Sj) = || e-^xY 
(r+l. s + l) 
2 (-1 )'[>,/ w WPIrt 1, s) -2VB*(r, s) 
k= 1 
+ /x 21 1/s B* (r, s +1)} - 4 (m# 2 )V0] 2 %) B ft (r, s)], 112 dxdy. 
(4*10) /n (nsj) = J/i/j 2 1| e~ {xi+,/) x 2 y 2 
i(r + 2, s+2) r 
2 <pn 2k (y) B 2A (r+2,s) + fB 2A (r+l,s+l) + B 2A (r,s + 2) 
k = 1 
• + 4 ^ 2 lff+1 ( B2 " +1 + 1 ’ ^ + BM + 1 ( r » S + 1 )) 
- (w* (r +1, s) + B 2A (r, s +1) 
2 h 
+ 
4&+1 
B 2A_ i(r+1, s) + B 2A_ i(7% 5+1) 
4 + J (2& + 2) (2/g+1 ) B s*+ 2 / X 
+ ^ t(4& + 3) (4&+1) ^ 
( (2^+1 ) 2 , (2£) 2 
\(4& + 3) (4&+1) (4& +1) (4&—l) 
+ r ^ 2Bl )-- B a - 2 (r, s) 
(4&+l)(4&-l) v ’ ’ 
+ 
+ 1 B 2A (r, s) 
-2 
2&+1 
4&+1 
BM+1 + s) + j|r- 1 B“- 1 (ns) 
dx dy. 
