384 
F. B. Wiley and G. A. Bliss 
and from [5] 
A^=fi3: + iy') =/{«)■ 
OX 
Thus 
dF (z) _ .dF (z) 
— ^ — - — ■ — z j 
dy dx 
from which we see that the Cauchy-Riemann differential equa- 
tions are satisfied and F {z) is monogenic. We conclude that 
There exists a function F (z) well defined and single valued in 
the rectangle E and such that 
./w 
for every z in E. 
We next take up a second auxiliary theorem: 
If in a neighborhood of a rectifiable curve C the function f (z) 
is single valued and continuous and the derivative of a single valued 
continuous function F {z), then 
Cfiz) dz = F(Z}~F(zo)^ 
JzoC 
where Zq and Z are the end points of C (Fig. 2). 
For use in this theorem we shall understand that 
r / ( 2 ) da = L 2 / (fi) 
Azi 
Az=0 k=l 
