76 Transactions of the Academy of Science of St. Louis 
chain of continuations of fi (x). 
Proof. Let f;(x) be a power series obtained by continuation 
from fi(x), necessarily in a finite number of steps. Denoting 
the range of f;(x) by R;:, we have 
f(x) = fil), wei, 
Now beginning with f;(x) as an element we can reverse the proc- 
ess, and obtain f(x), and consequently all the continuations of 
fi(x) and their subsequent continuations. This proves the 
theorem. 
We may now call any element f;(x), as well as fi(x), of the 
aggregate defining f(x) a generating element of the extended 
regular function. 
T.37. If the range of an extended regular function is a sim- 
ply-connected region R the function is single-valued every- 
where in R. 
Proof. It is sufficient to show that if we continue an ele- 
ment f(x) along two different routes between any two points Xo 
and x; we obtain the same value for f(x) at the terminus. 
Now since R is simply-connected we can replace one chain 
running from x» to x; by a succession of chains such that the 
neighborhoods of each contain the route of the previous one. 
It follows from an application of the uniqueness theorem (Corol- 
lary I to T.32) that with each succeeding chain we arrive at %1 
with the same value of f(x:), and since we can pass from one of 
our chains to the other in a finite number of steps the theorem 
is proved. 
T.38. A necessary and sufficient condition that f(x) be an 
extended regular function is that it be differentiable in a con- 
nected region R. 
Proof. The theorem is a consequence of T.25 and the defini- 
tion of an extended regular function by continuation. 
It is similarly possible to extend other properties of power 
series throughout a connected region obtained by continuation. 
Thus from T.22 we get 
T.39. Let f(x) be an extended regular function in a region R, 
which vanishes on an infinite sequence of points such as is de- 
scribed in T.22. Then f(x) =0 throughout R. 
Singularities. lf a function f(x) is regular on a sequence of 
points which has xo say, for a limit point, but is not regular at Xo 
itself, it is said to be singular at xo, and the point is called a 
singulzrity of the function. With this terminology we can re- 
