388 
F. B. Wiley and G. A. Bliss 
the difference between the integral from Zj, to taken along 
the side of the polygon K, and the term mentioned above will be 
L ^ / Ai,k) ~ ^i,k) ~ J fe) (^k+\ ~ 
71/.= 00 i 
= L ^ [/ (Zi. k) ~f (Zk) ] (Zi+1, k - Zi, t) . 
nj.= 00 i 
On account of the uniform continuity of / (z) in the neighborhood 
of C specified in the theorem, there exists, for a given e, a 5 such 
that 
I < e/L 
whenever z' , z" are in the neighborhood of C specified in the 
theorem and 
< 5 . 
The constant L is the length of the curve. Take 
Then 
and hence 
I j < § 
a' = (). . 
\ Zi.k -Zk\< j Zk+i ~ Zk \ < s. 
I / (Zi,k)-f (Zk) I < T 
It follows that 
Z 
^i,k) 
Summing these inequalities for /v = 0, . . . . , n — 1, and taking the 
limit as the numbers nk simultaneously approach infinity, we see 
that inequality [8] follows without difficulty, and we are able to 
conclude that 
k) dz = F{Z)-F(zo), 
which establishes our auxiliary theorem. 
From these two auxiliary theorems we see that the main 
theorem stated at the beginning of this section follows at once 
for a closed curve C when we recall that the function defined in 
the first of these theorems is single valued. 
