544 
MATHEMATICS: W, E. MILNE 
some respects more precise than those obtained by Birkhoff, and 
also to present similar asymptotic formulas for the n functions 
yujiy ' ' ' , yn, related to the n by the n identities 
These functions y play an important role in the theory of linear differ- 
ential equations. As is well known they form a system of linearly 
independent solutions of the equation adjoint to equation (1), while 
the expression 
Y,yi{x)yi{t) 
is of fundamental importance in Lagrange's method of solving the non- 
homogeneous equation of which (1) is the reduced equation, as well as 
in the formation of the Green's function of the system (1) and (2). 
Asymptotic forms for the ^''s were also used by Birkhoff in his paper on 
expansion problems. ^ 
I was led to make refinements in the forms of the ;y's and the y's in 
connection with a paper treating the degree of convergence of the ex- 
pansion associated with the differential system (1) and (2). 
Birkhoff divided the plane of the complex parameter p into 2n equal 
sectors, 
Sk : kir/n^d^Tg pSik-^-Vjir/n, ^ = 0,1, . . . ,2/^-1, 
and then numbered the n n-th. roots of — 1, Wi, W2, . . . , Wn, in such a 
manner that when p is on any given sector Sk, the inequalities 
R (pwi) ^ R (pW2) ^ . . . ^R (pw„) 
are satisfied, where R (pWi) denotes the real part of pWi. He then proved 
that if the coefficients Ps (x) in (1) have continuous derivatives of all 
orders in the closed interval a^x^b, there exist for p in any given 
Sk n independent solutions of (1), 
y, = w,(x,p)+e^^»^*-^^£,o/p-+\ 
yf^ = (x, p) e^^ E,,/p--^'-\ (4) 
^ = 1,2, . . . ,w; ^ = 1, 2, . . . ,w - 1, 
in which 
