Mathematics. — "On Elementary Surfaces of the third order'. 

 (First communication). Bj B. P. Haalmeijer. (Commimicated 

 by Prof. Brouwer). 



(Communicated in the meeting of May 26, 1917). 



Introduction. The existence of certain numbers of real straight 

 lines on cubic surfaces is well known. In Math. Ann. 76 C. Juel 

 makes a clever attempt to prove the existence of straight lines on 

 certain surfaces of the third order which are non-analytically defined 

 and which he calls elementary surfaces. His methods however are 

 not always convincing and some conditions he puts to his surfaces 

 seem to be artificial and out of place. The object of this note is to 

 introduce elementary surfaces of the third order in a natural way and 

 to prove the existence of at least one straight line on such a surface. 

 Our starting point is formed by the elementary curves of the third 

 order which are extensively dealt with by Juel in the Proc. of the 

 R. Acad, of Denmark, 7»»' series, t. 11 N". 2. Besides this we shall 

 principally use well known theorems of the analysis situs and the 

 theory of sets of points. 



In carrying out the following researches I am indebted for many 

 suggestions to Prof. L. E. J. Brouwer, who also has attracted my 

 attention to this subject. 



Definitions and e.vposition of the problem. An open Jordan curve, 

 which, together with the linesegment ') between its endpoints, forms 

 the boundary of a convex region, h qbWq^ convex arch. These convex 

 arches form the building material for the elementary curves. Let a 

 set of points be composed of a finite number of convex arches, in such 

 a way that it forms the continuous representation of a circle. To 

 every* point of the circle is to correspond one and only one point 

 of the set under consideration. Besides, the tangent {touching line, 

 Stiitze) is to change continuously with the corresponding point of 

 the circle and lastly the set of points is not to contain linesegments, 

 but may include entire lines. A closed set of points consisting of 

 a finite* or countably infinite number of these above defined sets 

 is called elementary curve. Isolated points are admitted though 

 tangents in the ordinary sense disappear. 



1) In the following line will be used for straight line. 



