( 142 ) 



Of this figure 1 we shall now construct a generalization ; in the 

 white band determined by the first n supporting threads we shall 

 namely di-aw the [n -\- If^^ supporting thread not tkrowjh the middle; 

 thereby each supporting thread segment gets an arbitrary distance 

 from the corresponding white band edge segments; however we 

 take care lirstly that the new supporting thread penetrates into each 

 segment of the corresponding w^iite band, and secondly that each 

 v2rtical supporting thread segment cuts the line /. 



In the more general figure it may happen that some segments of 

 coherence threads have expanded to bands, so that for this figure 

 we shall replace the name of coherence threads by coherence strips. 



And if each point of F which can be joined to the skeleton by 

 a line segment meeting the skeleton only in its endpoint, is added to 

 the skeleton, then also in the skeleton, just as in the coherence threads, 

 certain segments may expand to bands, so that for the new skeleton 

 we shall replace the name of supporting threada by supportiih/ strips. 



In the more general figure we assign to the points and inter- 

 vals in whicli / is cut by the n*^ supporting strip as their coor- 



2^ + 1 , . , . , 

 dinates the same numbers ^,, which in figure 1 appeared as the 



abscissae of the corresponding points of intersection of / with the 

 7i''i supporting thread, and each point or interval determined on / 

 by a coherence strip gets as its coordinate the number corresponding 

 to the Schnitt determined in the coordinates belonging to the supporting 

 strips. Then along / the coordinate is a nowhere decreasing continuous 

 function of the abscis, and like the abscis it has the initial value 

 and the endvalue 1. 



Now for a moment we abstract from the figure, and set apart a 

 finite or a denumerable infinite system of directly coherent sets of 

 numbers. The numbers belonging to these sets we shall call special 

 nunibers and we determine a coordinate function of the just now 

 described kind possessing each special numerical value over a certain 

 interval of abscissae, but each other numerical value (0 and 1 included) 

 only for a single abscis. 



Our aim is to construct the generalized figure 1 in such a way 

 that the coherence strips corresponding to the special directly coherent 

 sets of numbers get everywhere a finite breadth, whilst all segments 

 of the other coherence strips and of the skeleton get a breadth zero. 

 Starting from the coordinate function just now constructed on / we 

 succeed in this in the following manner: 



The first supporting thread we construct through the point of / with 



