Subdividing the Interior of a Rectifiable Curve 
387 
[ 8 ] 
{ f(z)dz- V / (z„) 
< e, 
where the sum is now taken along the curve Cj can always be 
made to hold for an arbitrarily chosen e by taking A;2k suffi- 
ciently small. 
The polygon K now is formed by joining in order^ the points 
of subdivision Zj, {k = 0, . . . . ,n) of the arc ZqZ of C (Fig. 3). On 
account of continuity and therefore the uniform continuity of 
<P {t)y f (t) on the interval Iq ^ t ^ T, Az can be taken small 
enough to insure that all of the points of K are in that neighbor- 
hood of C where / (z) and F {z) are single valued as indicated in 
the hypothesis of our theorem. In the expression [8] the term 
fizk) (Zk+I-Zt) 
corresponding to the element z^Zk-^i is replaced in the integral 
along K by the expression 
nj^-l 
T i ~ ^i,k) 
nk^ 00 i=Q 
where Zi^j, are points of subdivision of the line z^Zk^^o Since 
Zk+i - ~ 
