( 
Mathematics. — “Continuous one-one transformations of surfaces in 
themselves.” (2 nd communication). 1 ) Bj Dr. L. E. J. Brocwer. 
(Communicated by Prof. D. J. Korteweg.? 
(Communicated in the meeting of June 26, 1909). 
We shall now consider an arbitrary twosided 8 ) surface and we 
shall submit it to an arbitrary continuous one-one transformation in 
itself with invariant indicatrix. 
Under a limit region of the transformation we shall understand a 
region of the surface lying entirely outside its image region, but 
losing that property by any extension. 
However it may happen that a limit region allows of enlargement , 
i.e. can be united after an indefinitely small -modification of its 
boundary with a finite adjacent region into a new limit region, 
whose surface, measured by a certain system of coordinates, is then 
of course greater than the old one’s. This will be illustrated by the 
following developments. 
Under a transformation domain we shall understand a limit region 
not capable of enlargement, and our intention is to construct such 
a transformation domain. 
To this end we start from two arcs of simple curve s ), which are 
each other’s image, which have two and not more than two points 
in common, and which do not cross each other in those points. We 
shall suppose, that these arcs have no endpoint in common; their 
situation with respect to each other then still allows of various 
possibilities, indicated by fig. 1. 
With the aid of these two arcs we now construct two regions 
G and G', bounded by simple closed curves, which regions are 
each other’s image and lie entirely outside each other, whilst their 
bpundaries have two arcs of simple Curve in common. In fig. 2 
tips has been executed for the second possibility indicated by fig- i- 4 ) 
*) See these Proceedings , Yol. XI, page 788. 
2 ) A onesided surface falls under our result only when brought into a continuous 
multivalent correspondence with a twosided one. 
3 > i.e. “einfache Kurvenbogen” after Schonflies. In my preceding communications 
of this year (these Proceedings XI, p. 788, 850) 1 translated “einfach” by “single . 
Finding the term “simple closed curve” used by Veblen, I shall adopt in*future 
tips mode of expression. 
4 > 0nl y in the fifth and eighth case of fig. 1 this might give rise to some diffi¬ 
culty, namely if the two common points of the arcs are each other’s image, y 
a slight modification of the figure this difficulty can then be cancelled. 
