Subdividing the Interioi' of a Rectifiable Curve 377 
points can be joined by a rectifiable curve every point of which is 
an interior point, and a similar statement holds for exterior points. 
Any continuous curve joining an interior point and an exterior 
point must have on it at least one point of the curve C. Every 
point of C is a limit point of both interior and exterior points. 
We define, for the moment, the effective length of a curve to be 
the length of that part of the curve that lies in no horizontal line. 
We state the auxiliary theorem; 
If yi and are the maximum and minimum values of y in the 
interval 
t' ^t ^ t" 
for a simply closed rectifiable curve C 
X = ip{t), y = I(t) C' ^t S t") 
then there is a segment p'p" of the horizontal line I, 
y = Kj/i + j/2)> 
interior to C except for its end points, which forms ivith C two simply 
closed rectifiable curves. If 
- ^2 > e, 
the segment can be so introduced that each of these curves has an 
effective length greater than e. 
Let Pi and po be the two points on C at which y is a maximum 
and a minimum respectively. Select points pf and pf which are 
interior points of C and so near to pi and p 2 , respectively, that the 
former is above the line I and the latter below it. We may join 
the points p\ pf by means of a continuous polygen D having a 
finite number of sides and consisting entirely of interior points 
of C. 
Any side of D which has an end point in common with the line 
I may be rotated slightly about its other end point, and in this way 
it may be brought about that D has only interior points of its 
sides in the line I, and actually crosses the line where they have a 
point in common. 
The polygon D must intersect I at least once, say at a point p, 
since one end point of D is above and the other is below the line. 
There will be a segment p'p" of I containing p such that p' and 
p" are on the curve C while every other point of the segment is 
interior to C. There can be only a finite number of such seg- 
ments p'p" since D has at most a finite number of intersections 
with the horizontal line. There must be at least one of them on 
