646 MATHEMATICS: E. B. STOUFFER Proc. N. A. S. 
When the coefficients of (1) are subject to the conditions (2) it is easily 
proved by induction that 
where 
n 
Qikr = Qi,k,T—l + ^^i Qi^.T— iQxkl. 
X=l 
If Tiki denotes the coefficient of {B) corresponding to pi^i of (A), we find 
by straight substitution 
24 
X=l 
n m — X — / 
2 2 ('^ /) ' + y = 0,1, - 2), (4) 
where A^i is the algebraic minor of ay,i in A. By the use of (3) the expres- 
sion (4) for TTiki may be put into the form 
n n 
X=l M=l 
where 
n m — 'S. — l 
_ ^ ^ (m — l\ 
f=l 7=0 
Again, we find by differentiation 
n n 
X=l M=l 
where 
Similarly 
where 
^=1 
X=l u=l 
^X 
