MATHEMATICS: W. E. MILNE 
545 
where m is any positive integer or zero.^ The En are functions of x 
and p which are bounded for x in (a, h) and for p in Sk and large in 
absolute value. The ji are analytic in p in Sk, and the {x) have 
derivatives of all orders with respect to it: in (<z, h). 
The modification here proposed is this: // the coefficients Ps {x) have 
continuous derivatives of order {m -\- n — s) in (a,b), m being any positive 
integer or zero, then there exist n solutions of (i) of the form (4), ana- 
lytic in p in the sector Sk, and the functions Ui {x,p) are of the form 
u,{x,p) =e^i''-'^[l-h<Pi(x)/pw,-{- . . . +^^ix)/{pwri 
where the functions <pj{x) have continuous derivatives of order (m + n—j), 
and are independent of i. 
The improvement in precision over Birkhoff's formulas consists 
primarily in putting the w,: (x,p) in the form indicated, where the <pj (x) are 
independent of i; the details concerning the number of derivatives of 
the F^s and the ^^s are of secondary importance. A similar remark 
applies to the statement concerning the y^s, which is as follows: 
Then functions yi, determined hy the n equations (J), have when \p\ is 
large the asymptotic form 
_ -pwi(x-c) _ 
yi= % ._, [vi(x,p)+£i/p'^^% i^l,2, . . , ,n, 
ivhere 
Vi{x,p) ^\^-^Pl{x)/pWi-{- . . . -h rp^ix)/{pWiy\ 
in which the functions \pj {x) are independent of i and have continuous 
(m n — j)-th derivatives in {a, h). 
The proof of the asymptotic formulas for the ;y's is simply an adap- 
tation of the proof given by Birkhoff . In the case of the 3'^s, we verify 
the formulas by substituting the values of the yi given above into equa- 
tions (3) and showing that the t^'s and E's with the desired prop- 
erties can be chosen to satisfy them. 
1 Trans, Amer. Math. Soc, 9, 219-231, 373-395 (1908). 
'Loc. cit., p. 391, formula (56). 
•For the formulas here quoted see loc. cit., pp. 381-2, formulas (21) and (23). They 
are quoted in (4) with some slight changes 
