Subdividing the Interior of a Rectifiable Curve 
389 
§4. The Cauchy Theorem for a Curve C in any 
. Simply Connected Continuum 
In this section we take up the proof of the main theorem of our 
paper. 
Subdivide the region interior to the simply closed rectifiable 
curve C lying in a simply connected continuum until each subdi- 
vision can be surrounded by a rectangle lying wholly in the 
continuum (see §1). The integral of f{z) taken around the 
border of each sub-region exists and has the value zero ( §3) . The 
integral of/(^) taken around the curve C is equal to the sum of the 
integrals off(z) taken around the border of each of the sub-regions 
in the sense determined by the direction the integral is taken 
around C. Thus we have the theorem : 
The definite integral 
J/ 
exists and has the value zero if the path of integration C is a simply 
closed rectifiable curve lying within a simply connected continuum on 
which the integrand function f{z) is single valued, continuous, and 
possesses a single valued derivative. 
The University of Chicago, April, 1913. 
