( 1-n ) 



Our starting-point is figure 2 (Plate I) explained i.e. p. 423 and 

 424 of Vol. 68 of the Mathem. Annalen, wliicdi figure in the following 

 will be called the primitive Jigure. 



We tirst simplify the primitive figure by leaving out the red band, 

 and by reducing the breadth of the black bands to zero. These 

 contracted black bands we shall call ''supportimj threads', and we 

 draw each of them through the middle of the white band determined 

 by all the i)receding supporting threads, as is executed here in figure 1 

 for the first four supporting threads (this figure is to be looked at 

 in the position indicated by the subscription : Fig. 1). 



The rectangular circumference of figure 1 we shall indicate by k, 

 the circumference together with its inner domain by i'"'. The circum- 

 ference together with the supporting threads we shall call the .9l-^/t^/o« 

 of the figure. Two arbitrary points of the skeleton possess the pro- 

 perty of being contained in a perfect coherent part of the skeleton. 



We now consider a horizontal section / of figure 1 cutting all the 

 vertical line segments of the supporting threads, and we determine 

 the points of / by their ^(^.3^/5, i. e. their distance from the left endpoint 

 of /. The length of / we choose as unity of length. 



Then the abscis of the point of intersection of / with the first 



1 

 supporting thread is — ; the abscissae of the points of intersection of 



u 



1 3 



/ with the second supporting thread are — and — ; those with the 



4 4 



13 5 7 



third supporting thread are — , — , — and — ; and so on, 



8 8 8 8 



So the set of points determined* on / by the system of supporting 

 threads possesses as their abscissae the set of dual fractions between 

 and 1. 



Two points of F will be called directly coherent, if they are con- 

 tained in a perfect coherent part of F having no point in common 

 with the skeleton. Two points directly coherent with a third point 

 are also directly coherent with each other. The points directly cohe- 

 rent with a given point form a set which will be called a cohe- 

 rence thread. 



The abscissae of the points of intersection of / with a coherence 

 thread form a set of numbers to be called a directly coherent set of 

 nunihers. Two abscissae then and only then belong to the same 

 directly coherent set of numbers, if either their sum or their diffe- 

 rence is a dual fraction. 



The set of coherence threads jiossesses the po\ver of infinity of 

 the continuum. 



10 



Proceedings Royal Acad. Amsterdam. Vol. XIV, 



