MATHEMATICS: G. D. BIRKHOFF 
579 
of classification since the solutions are carried over into the solutions 
of equivalent differential equations by such transformations.^ The 
precise statement of the result is contained in the following 
Theorem. Let Y (oc) be a matrix'^ of functions (x) {i,j = 1,2, - , n)y 
each analytic for \ x \ '^r, with determinant Y (x) not zero for \ x\^r. 
A necessary and sufficient condition that Y {x) forms the matrix solution 
of a linear diffierential system 
dx 
where the elements a,-j (x) of A (x) are analytic (q = 0) or have a pole of at 
most the qth order (q>0) at x = co ^ is that, for every function cp (x) 
of the form 
<p (x) = x-\-l x~^ + m X + • • • , 
the matrix M {x) defined by the matrix equation 
Y {cp{x)) =M(x) Yix) 
is composed of elements mij (x) analytic at x= co , 
Let us prove first that the condition is necessary. 
The function M{x) defined for any choice of <^(x) is single-valued as 
well as analytic for |x| sufficiently large. In fact, when x makes a cir- 
cuit of x= oo , F(x) is altered to Y{x)C where C is a matrix of constants 
of determinant not zero. But since (p{x) —x remains finite as x becomes 
infinite, (x) will make essentially the same circuit if \x\ be sufficiently 
great. Thus F(<p(x) ) changes to Y{ip{x) )C, and M{x) is unaltered in 
value. 
Now the expression for M{x) 
Y (xo) Y-^ (xj (xq = <p (x); x^ = x) 
may also be written as a matrix product 
[Y{xo) (x,)] [Y (xi) fe)] • • • [F F-' (*J]. 
Here we will assume that the values :^o, Xi, X2, . . . , Xm of x are 
equally spaced, each after the first differing from its predecessor by 
Ax= {xi — Xm)/m. 
From the differential equation itself, it is clear that 
F fe) - F (xi+i) = Ax[A (xi) + Ei] Y (xi^{) {i = 0 zero, 1, • • • , w - 1) 
where Ei will have arbitrarily small elements if Ax is small enough. 
