&4 Transactions of the Academy of Science of St. Louis 
Let us suppose that the set of equations (8) have been solved. 
They are identical in form with those given, say, in Theory of 
Infinite Series by Bromwitch, p. 156, 2nd ed., in the inversion 
of complex power series, where it is shown that the inverted 
power series has a positive radius of convergence. Hence by 
the correspondence principle the same, at least, is true of (7). 
The result may be stated as in the following theorem. 
T.43. If f(x) is a regular function on Ey, to Ej such that all 
its derivatives at x =x» form a class of mutually multiplicable 
vectors, and such that f’(xo) is not a singular point of fy (x) ee 
then there exists an inverse function which is regular in a neigh- 
borhood of xp. 
Differential Equations. First, it is seen that the extension 
of the notion of regularity to functions of several variables can 
be easily made. Thus, we say that f(x, y) is regular in the 
(p1, p2) neighborhood of (xo, yo) if 
f(x, y) = ae Btrintm|(x% — xo)"(y — Yo)™, 
and if the series converges uniformly for || x —xol| <pi<0 and 
lly — yol| <p2>0. The general properties of regular functions 
may be carried over; in particular, that of continued differentia- 
bility. 
We consider the differential equation 
— = fey) 
4 
where f(x, y) is a regular function on Ey? to Ey) in a neighbor- 
hood of (xo, yo). If the coefficients in the representative power 
series are multiplicable among themselves, it may then be as- 
serted that the differential equation admits a unique solution 
y=y(x) on Eqy to jy, which is regular in a neighborhood of xo, 
and which reduces to yp for x =x». The method of proof is sim- 
ple. We assume a Taylor’s series expression for y(x), and with 
the aid of T.13 and T.14 we calculate the successive derivatives 
of y(x). For example, we have for y’’(x) 
eo 
y’"(x) = >, Ofrentm) | 2(x — xo)™ y — Yo)™ 
8 
+ m(x — xo)"(¥ — Yo) m—Ly’( a) } 
