( 300 ) 



Mathematics. — "Continuous one-one irans/ormations of surfaces 

 In themselves." (4^^^ communication ^)). By Dr. L. E. J. Brouwer. 

 (Communicated by Prof. D. J. Korteweg). 



(Communicated in the meeting of May 27, 1911). 



In this communication as in the preceding one we shall occupy 

 ourselves with continuous one-one transformations with invariant 

 indicatrix of a two-sided surface in itself. 



If for such a transformation there is an invariant arc of simple 

 curve, it contains at least 6>?ie invariant point ; more than one invariant 

 point need not appear. 



If, however, each of its two sides is invariant, then tiie arc contains 

 at least t>ro invariant points; more than two invariant points need 

 not a})pear. 



Of the former of these two evideni theorems we have sliown in 

 § 2 of the third communication that it can be extended to the most 

 general circular continuum (of which the arc of simple curve can 

 be regarded as the simplest type) ; to the latter theorem we shall 

 give the same extension in the following. 



A segment of the circumference formed by the accessible points 

 of a circular continuum will be called a complete circumference 

 segment, if the set of its limiting points is identical to the circular 

 continuum itself. 



As the generalization of the arc of simple curve with two invariant 

 sides we can consider a circular continuum (p' whose circumference 

 can be divided by two "Schnitte" into two complete circumference 

 segments, both invariant for the transformation. 



Of (f' together with a certain vicinity i|/ we construct a continuous 

 one-one representation on a finite region of a Cartesian plane, where 

 they pass successively into (f and if'> ^^nd we draw in that Cartesian 

 plane a simple closed curve x lying together with its image and its 

 connterimage in if', whilst its inner domain contains <f . 



All figures to be constructed in the following and likewise their 

 images and their counterimages we suppose to lie in tp. 



According to the third communication 7 possesses a point /invariant 

 for the transformation; we shall suppose that this point /is the only 

 invariant point of (f. 



The two Schnitte determining on (f the two invariant complete 

 circumference segments 0^ and 0^ , we shall represent by S^ and S^ . 



1) See these Proceedings Vol. XI, p. 788, Vol. XII, p. 286, Vol. XIII, p. 767. 



