A METHOD OF SUBDIVIDING THE INTERIOR OF 
A SIMPLY CLOSED RECTIFIABLE CURVE 
WITH AN APPLICATION TO 
CAUCHY’S THEOREM 
F. B. Wiley and G. A. Bliss 
The Jordan^ proof of the Cauchy theorem requires that all 
points of the closed curve C and its interior lie in a region in 
which the integrand function is continuous and has a continuous 
derivative. In the Goursat and in the Moore proofs^ the require- 
ment that the derivative be continuous is avoided, but there are 
still restrictions on the character of the curve C other than that 
it be rectifiable. These restrictions are indicated in Moore’s 
statement of the theorem which we quote: 
‘^The definite integral 
exists and has the value zero, if 
(1) the path of the integration C is a simply closed continuous recti- 
fiable curve met by the various lines parallel to the xy-axes in the 
2 '-plane {z = x-\- iy) in a finite number of points and segments of coin- 
cidence, and moreover having the property (2) ; 
(2) for every point t of C, if a square with sides parallel to the axes 
converges in any way to the point the ratio of the total length of the 
arcs of C lying on the square to the perimeter of the square is ultimately 
less than a certain constant which maj^ vary as ^ traverses C] 
(3) on the region R, consisting of the curve C and its interior region, 
the integrand function / {z) is a single valued continuous function of 2 
with a single valued derivative/' {z)R 
The restrictions of (1) and (2) on the path of integration C, 
otherwise than that it be closed and rectifiable, have been avoided 
by Moore at the close of the article to which we have referred, by 
1 Jordan, Cours d’ Analyse, 2d. ed., vol. 1, (1893), §§196-198. 
2 Goursat, Cours d’ Analyse Mathematique, vol. 2, (1911), §§286-7. ]Moore, 
Transactions of the American Mathematical Society, vol. 1, (1900) p. 499. 
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