82 Transactions of the Academy of Science of St. Lows 
(5) 2 Sear 2; Birtnj x” 
0 
where the coefficients are undetermined and 7 is appropriately 
chosen. This is substituted into the defining equation, whic 
may then be reduced to a power series, equated to zero, and 
whose coefficients are, therefore, equivalent to nullifying ele- 
ments. In general this yields an infinite set of recurrence for- 
mulas which are sufficient to determine the coefficients of (5) in 
terms of those that are known. It only remains to show that 
the radius of uniform convergence of (5) is greater than zero, 
and it is here that the principle of correspondence plays its most 
important role; for since the defining equation is made up of 
regular functions, which we assume to be finite in number, or 
else reducible to a finite set, the unknown function y is of the 
type (3). But on account of property II we are assured that 
whenever the corresponding function (4) exists in some neigh- 
borhood of a given point the same is true of (3). Thus, in order 
to derive theorems on implicit functions for the abstract varia- 
ble, it is only necessary to refer to the analogous theorems in the 
theory of the complex variable, and whenever the process there 
is such a one as described above, involving only addition, multi- 
plication, differentiation, and integration, the theorem for the 
abstract variable is equally valid. 
The problem of inversion is of fundamental interest in the 
theory of implicit functions. Suppose we are given a power 
series Do @[r4n)x" and the equation 
(6) y= > Oise”. ; 
0 
The problem of the functional dependence of x on y may be 
solved under certain conditions by assuming this dependence 
to be expressed by an equation 
(7) oe a pas Bir+mjy™ 
0 
where b;,-:m) is an undetermined coefficient. Assuming xeZ111 
the only value of 7 which will satisfy (6) and (7) is 1, so that the 
functions considered are on Ep to Epj. Without restricting 
the general case, we can set @p; =0, and, consequently, b,:; ~ 9- 
_ We have, on substitution 
