MATHEMATICS: G. D. BIRKHOFF 581 
we observe that M{x,p) is made up of elements single-valued and ana- 
lytic in X and p if 
kl^i + f, IpI^I, 
inasmuch as in this event 
Moreover, in view of the stated condition, the elements m%j(x,p) 
of M(x,p) are analytic in x at x = oo . 
Consider now the integral formula 
i^,fep) = -5^ r??j|^ (|.|>l + r) 
X 27rV-l ^ U^-^) 
where the integration is performed around the fixed circle C: \^\ = l-\-r 
in a positive sense. This reduces to the Cauchy integral formula by 
the substitution x=l/x\ which holds since m{x,p) is analytic 
for X ^ 1+r and at x= oo. 
We now perceive from the form of the integrand that ma (x,p) must 
be analytic in x and p for x = oo . 
Differentiating the equation of definition for M{x,p) with respect 
to p we obtain 
x-'4- Yi^ + P^"') = (x, p) Yix). 
dx dp 
The elements of dM{x,p)/dp are also analytic in x and p at x= co . Let- 
ting p approach zero we obtain, for |xl >l-\-r, 
x-'^-^ = ~M(x,0)Y(x) 
dx dp 
which establishes at once that Y{x) is the solution of a differential 
system of the desired form. 
^ This result was partly known to me in 1908. For a special case, see Trans. Amer. Math, 
Soc, 14 462-476 (1913), in particular pp. 475-476. 
' For the elements of the theory of matrices here used see Schlesinger, Vorlesungen iiber 
linearen Differentialgleichungen, pp. 18-19. 
