('146 y 



Proof. Let AT be a point taken arbitrarily in the inner region 

 of C, and let a^, a^, a^, . . . . be a fundamental series of indefinitely 

 decreasing arcs of simple curve lying abroad from their end|)oints 

 in the inner region of C, whilst the endpoints belong to c^liy ^^nd 

 are separated by the Schnitt s. In this series is contained a series 



b^, b.^, bs, converging to a single point P of o, and in which 



each bn is separated from M by C -\- è„_). We can then join M 

 and P by an arc of simple curve z cutting an intinite number of 

 tlie by, in the order of their indices each in one point and passing 

 there from their side turned to M to their side turned to s. The 



by. intersected in this way form a series cl^, d^, c/,, Let z,i be 



the part of z enclosed between (/„ and r/„_|_!, 7)„ the point of inter- 

 section of z and d,„ An resp. Bn the left resp. right endpoint of 

 d„, (fn resp. \pn the arc of curve determined on C by the Schnitte 

 corresponding to A„ and .4„_^i resp. to /i„ and ^„-^i, (>„ resp. t„ the 

 part of the inner region of C cut off by the arc of simple curve 

 Ay: D„ I),i-\-] A,.tj^\ resp. B,, D^ D,i-\-\ Bn-[.\, Un resp. v„ the part of 

 <fn resp. V^« b'i"o i^ ^») resp. Qn- We then can join Z)„ and Z)^-^i by 

 an arc of simple curve /„ lying entirely in the part of the inner 

 region of C enclosed between (/„ and d,i-\-\ and moving away from 

 ^^^ _|_ ji^ _[- n^^ no farther than a certain maximum distance 6„ inde- 

 finitely decreasing for indefinitely increasing n. .These arcs tn form 

 together an arc of curve possessing the properties required. 



Theorem 4. If the closed curve C is divided by die Schnitte s^ 

 and s.^ of r'^]- into two proper (i.e. not identical to C) arcs of curve 

 C and C.^, then the points- common to C, and C^ form a non-coherent 

 set of points C\^. 



Proof. Let ^i resp. o.^ be the juncture belonging to s^ resp. s^, 

 then according to tlie lemma just now proved we can draw from 

 a point M taken arbitrarily in the inner region of C to ends ^^ and 

 e^ contained in o^ resp. o^ two arcs of curve which abroad from 

 their ends are simple, do not meet each other, and lie entirely in 

 the inner region of C. These arcs of curve we represent by f^ and 

 F2 (see the schema in fig. 2), and the largest perfect coherent part 

 of C12, containing e^ resp. c^, by p, resp. p^. Let Q, resp. Q^ be a 

 point of cm belonging to C^ but not to C^, resp. to C\ but not to 

 C.. Then from M to Q^ and Q^ we can draw paths iv^ ajid la^ 

 which abroad from their ends lie entirely in the inner region of C, 

 and meet neither each other nor f^ or f^. In the inner region of 

 C these paths w^ and lu^ are separated by f^ and r^. 



About Qi as centre we describe a small circle y.^ which together 

 with its inner region has no point in common with C^ -\- f ^ -]- f ^, 



