580 MATHEMATICS: G, D. BIRKHOFF 
For 1^1 taken large enough, it is also clear that we will have an equality 
of the form 
\aij{x)\<a\x^\ = 2, - .n) 
for the elements of A (x) . Hence if I denotes the matrix with elements 
8ij{8ij = 0, i=pj; dii=l), and if A denotes the matrix whose elements are 
all equal to the positive constant a, we find that the elements of the 
matrix 
Y (Xi) fe+i) or I + Ax {A fc) + E,) 
are less in absolute value than the corresponding elements of the matrix 
I + 2\Ax\A\xi\', 
Also on account of the form of (p(x) we have for \x\ large enough 
I — it;^ I ^ ^ I x^- 1 {k independent of i or x), 
and thence 
\Ax\^~\xA-\ 
m 
Substituting this value oi Ax above, we conclude that the elements 
of F(x»)F~i(xt+i) are less in absolute value than the corresponding 
elements of 
m 
Thus, by the product formula, M{x) will have elements whose absolute 
value is less than that of the corresponding elements of 
(lJr—A\ 
\ m / 
whose elements are obviously limited in absolute value for all values of m. 
Consequently the elements of Mix) are limited in absolute value for 
|x I large enough, and are necessarily analytic at infinity. 
Let us show now that the stated condition is sufficient. 
To this end we will consider the family of functions ' 
<p {x) =x-\- px~^ 
where p is a parameter. 
Defining M{x,p) by the equation 
Y{x^-px-')=M(x,p)Y{x), 
