380 
F. B. Wiley and G. A. Bliss 
§2. The Cauchy Theorem for A Rectangle 
In the present section we show that the definite integral 
/ 
f / (z) dz 
Jr 
exists and has the value zero, where R is the border of a rectangle 
in the interior of, and on which, /(£;) is holomorphic. 
We take up at once the theorem of this section of the paper, 
using the method of proof due to Moore.® 
The definite integral 
exists and has the value zero if the path of integration R is any 
rectangle, and if in the interior of R and on R itself the integrand func- 
tion f{z) is a single valued continuous function of z with a single 
valued derivative f' (z) . 
We may consider without loss of generality the rectangle R as 
given with its sides parallel to the xy-axes. Since the path 
curve R is rectifiable and f{z) is continuous on R the integral./ 
exists.^ We show by indirect proof that J = 0. 
Set \ J \ = 7] and assume -q > 0. By the introduction of two 
diameters, the rectangle is subdivided into four equal rectangles 
T? ' T? " T? J? ff" 
ill J -Hi , ill , it] 
Define Jf and q by the equalities 
{ f (z) dz, 
Jr/ 
Ji\ = vf, 
and likewise for Jf", and /]"". Then we have 
J = // + //' + //" + /]"" 
and 
0 ^ ^ / I //I fff \ ffff 
< q ^qi + qi A q\ A q\ • 
Hence at least one of the four t^i’s must be at least q/d. Choose 
such an qi and denote it by qi without superscript. Do likewise 
® hoc. cit. 
^ Moore, loc. cit. 
