382 
F. B. Wileiy and G. A. Bliss 
where is the total length of the rectangle Rp. 
then 
X. ^4t2-^ 
7), g4 V2eT-4-‘'. 
This, because of [1], gives 
7 ] ^ 4 ^2 e 7“. 
Since 
But the positive number e remains at our disposal and, since tj 
is greater than zero, may now be so chosen that 
4 '^2 €7“ < 7 ]. 
This is in contradiction with the preceding inequality, and com- 
pletes the proof of the theorem. 
§3. The Cauchy Theokem for Any Simplf" Closed Recti- 
fiable Curve in a Rectangle in Which the Integrand 
Function Has a Derivative. 
In this section we establish the theorem: 
The definite integral 
Jy (z) dz 
exists and has the value zero if the path of integration C is any 
closed rectifiable curve consisting only of interior points of a rectan- 
gular region on which the integrand function f ( 2 :) is a single valued 
continuous fu7iction of z with a single valued derivative f' {z). 
We establish this theorem by setting up a single valued func- 
tion F {z) which has the derivative f (z), and then showing by 
means of an auxiliary theorem that for any rectifiable curve C 
in the rectangle 
F,(Z)~F(zo) = rf(z)dz, 
JzqC 
from which the proof of the main theorem follows. 
The integral 
ff (2) dz 
