Outline of a Theory of Functions of an Abstract Variable 81 
the form 
(1) 81(4ir], G4}, > + ; %) 
G2(Di01, Opep11, -- + ; #) 
and their corresponding power series 
(2) gi(\| apr eral, - ++ 38) 
go(|[Bte3||, |[oteenll,--- 38). 
It is supposed, naturally, that gi(---) and go( - - - ) do not nec- 
essarily lie in the same space. Now if a third power series h(x) 
is formed from gi(:--) and go(---) by means of the postu- 
lated operations of addition and multiplication among their ele- 
ments (this, of course, includes forming the derivative and in- 
tegral), so that h(x) may be written as 
(3) A( air), Girt}, "°° ; Ots1, Wistily 5 x) , 
the principle of correspondence asserts that by means of the 
analogous operations in the number system there exists a com- 
plex power series with positive real coefficients equal to the func- 
tion 
(4) h lag) 
which has the properties I and II, and which, therefore, corre- 
sponds to (3). The fact that I and II hold between (4) and (3), 
as well as between (2) and (1) is a direct consequence of the in- 
equalities that appear in P.12, P.13, and P.21. 
Implicit functions. The equation y=f(x) is said to give y 
explicitly in terms of x, supposing, of course, that f(x) isa clearly 
defined assertion that establishes the functional correspondence 
between x and y by means of operations upon x alone. Whena 
variable y is defined unambiguously as a function of x, but not 
in the explicit form y ~ f(x) it is said to be defined implicitly. 
The implicit functions which we consider are those which in- 
volve only the postulated operations of addition, multiplication, 
and the limit process, but more particularly in a TT — 
that the explicit functions which do appear in the defining rela- 
tions, which may be algebraic, differential, or integral equations, 
are all regular functions. In that case if the problem is to de- 
termine y as an explicit function of x in the neighborhood of 
some point, say the origin, we set 
, [Oreeull, >> 38) 
I) [lates], = ~~ 5 [tea] 
