Subdividing the Interior of a Rectifiable Curve 
383 
exists because the curve C is rectifiable and the function / {z) is 
continuous on CC We define R (Fig. 1) as a rectangle with its 
sides parallel to the x^-axes; (a, b) as the lower left-hand vertex 
of R] 2 : as any point (x, y) of the rectangle; L as the path from 
{a, b) to {x, y) that is parallel to the axis of reals from (a, b) to 
(x^ b) and then parallel to the axis of imaginaries from {x, b) to 
{Xj y); as the path that is parallel to the axis of imaginaries 
from (a, b) to (x, y) and then parallel to the axis of reals from 
(a, y) to (x, y ) ; and 
F{z)= fj{z)dz. 
From the definitions it follows that 
[4] F{z)= ^ f {x-Cib)dx-Ci ^ f {x iy) dij . 
Ja Jb 
In consequence of Cauchy’s theorem for a rectangle as proved 
in §2 this may be written 
[5] F{z)=i I f{a-\-iy)dyF ^ f(xFiy)dx. 
kJ b tJ a 
From [4] we have 
\^^^=f{x + iy) 
^ by 
Moore, loc. cit. 
