( 302 ) 



^5 bv a path < t\ |/2, whilst farthermore Os and R,i may be 



3 



connected with (f by paths Q^ Hi and Rs K,i <l^^r, , lying 



»'i I p t ^ I p ' p ^ ' fj 



outside '?>; , and not cutting 5^3 _, thus containing S^ between them. 



These tliree r)aths form a skeleton arc Ha Qs Rs Ks whose size 



* ' p ' p ' p ' p 



for indefinitely increasing /; converges to zero, and which we shall 

 represent by ^,5 . 



So out of the series of the (ip we can select a fundamental series 

 Tj, r,, . . , in such a way that for indefinitelij increasing p the skeleton 

 arc OfT converges to a single point V not identical to P. 



We shall now suppose that the Schnitt >S\ is determined by an 

 accessible point P. Let in that case iv be a path leading to P, and 

 let s^,s.,,. . . be a fundamental series of skeleton arcs separating S^ 

 from X, and whose size converges to zero. Then as soon as ]> has 

 exceeded a certain value, all s^, must cut iv, and that in points which 

 for indefinitely increasing p uniformly converge to P, so that 5^ con- 

 verges for indefinitely increasing p uniformly to P. 



So if 'S'l resp. S^ is not determined by an accessible point coin- 

 ciding with /, we can construct a skeleton arc U^ V^ resp. f/, V^ as 

 small as we like, separating S^ resp. S^ from y., and not cutting its 

 image U\V\ resp. U\V\, so that either the circumference segment 

 f7i Fi resp. U^V^ is a part of the circumference segment U\V\ 

 resp. U'.^V\, or the circumference segment L''i F'l resp. U\V\\'iSi 

 part of the circumference segment U^ V^ resp. f/, F,. 



Farthermore it is impossible that S^ and aS, are determined by 

 accessible points coinciding with each other, for, in that case (he 

 derived sets of 0^ and 0.;^ would have only that one point in common, 

 so that 6»! and 0^ would not be complete circumference segments. 



On 6>i we choose a point P not coinciding with /; the image of 

 P we represent by P\ the image of P' by P", the counterimage 

 of P by Pi. From y. we draw to P, P', P", Pi paths iv,z,u,v not 

 meeting each other, and containing such endsegments e, e', e", ei that 

 e' is the image of e, e" the image of e',ei the counterimage of t^ and 

 we construct an arc of simple curve k starting in P, not passing 

 through /, cutting 0^, and not meeting iv ; the image of i?; we repre- 

 sent by k', the image of k' by k", the counterimage of k by ki, the 

 size of k, k', k", k successively by g, g', g", gi, the largest resp. smallest 

 one of the latter four quantities by gh resp. gi. We describe circles 

 a, a', a", a; containing in their inner domains J,/, ƒ ',^8- at a distance 

 gh successively the arcs k,k',k",ki, and we take care to choose /.• 



