( 303 ) 



so small that two arbitrary ones of the sets of points iv-\-J, - +ƒ, 

 u-{-j", v-]-ji possess a distance ^8^;/j from eacli other, that the 

 parts of io,z,u,v contained in j,j',j",ji belong entirely to e,e',e",e;, 

 and that k cannot contain a skeleton arc separating a Schnitt S^ or 

 /Sj determined by an accessible point coinciding with /, from the infinite. 



Either k or k' contains a point Q of o^ accessible from y. along a 

 path not cutting cf -}- ^" + ^■'. In the following we shall assume Q 

 to belong to k ; if it were to belong to k', we might consider instead 

 of the given transformation its inverse, and then follow the reasoning 

 of the text. 



From z to Q we lay a path m not cutting rp -\- k -}- k' -\- lo. 



The part of k contained between P and Q we represent *by r, 

 its image by r', the image of r' by r". If we then approximate ^/ -\- r 

 at a sufticiently small distance by a polygon ^, this polygon P con- 

 tains two arcs p^ and p.^ both connecting to and m, and having no 

 point in common. Together with certain parts of lo -\- r -\- m these 

 arcs p^ and p.^ form two polygons ^^ and '^, whose inner domains 

 have no point in common, so that the inner domain of e.g. ^Pj does 

 not contain the point /. We then determine the positive sense of 

 circuit of the circumference of (f hj a circuit from Pto (^2 inside ^13i. 



The circumference segment PQ contains one and not more than 

 one of the two Schnitte Si_ and S^ : we may assume the Schnitt 

 aSi to belong to the circumference segment PQ. 



Then *Si cannot be determined by an accessible point coinciding 

 with /; for, in that case r conld not contain a skeleton arc separating 

 S^ from the infinite, so that tlie point / would be accessible inside 

 ^1, which is impossible, / lying outside ^^3,. 



We represent the image of Q by Q, and according to the manner 

 of succession of the points P,P', Q, Q' for a positive sense of circuit 

 we distinguish four cases. 



First case: P' precedes P, and Q' precedes Q. 



In this case r contains a skeleton arc d separating (2' from the 

 infinite, and accessible from the infinite without a crossing of 7 -[-/'-[- r'. 

 Let M be the endpoint of d preceding Q' on the circumference of 

 q), t a segment of d containing M, c the part of r thai i-emains after 

 destroying in r all skeleton arcs separating (2' from the infinite. 



Between the image ?o' of iv and t we construct a i)olygonal line 

 ^',, and between t and the image m' of ?h a polygonal line '^V^ which 

 both approximate q -\- c -\- r -\- r" at a distance e. 



The segment cut off from lü' resp. t by ^', we represent by 



