emch: on a special class of connected surfaces. 



55 



the boundary of the surface and divide this cross-cut into 2a — i equal 

 parts. At each of these division points let a loup-cut be drawn 

 parallel to the boundary of the surface; we obtain in this manner a — i 

 bifacial surfaces of double the length of the original surface, i. e. of 

 double the length of the middle line of the ribbon, and one unifacial 

 surface of the same length as this middle line. If the above men- 

 tioned cross-cut be divitled into 2a instead 2a~i equal parts and the 

 loup-cuts be drawn as before, we shall obtain 2a new surfaces all of 

 which are bifacial and all of length equal to the middle line of the 

 ribbon. 



Each loup-cut drawn parallel to the boundary of a unifacial sur- 

 face from the middle point of the cross-cut produces a bifacial surface 

 having double as many twists as the original. A loup-cut drawn in a 

 bifacial surface divides it into two other bifacial surfaces, each with 

 the same number of twists. If a loup-cut starts from the division 

 point a — I of the cross-cut, a unifacial surface is divided into a — i 

 bifacial surfaces, each having double as many twists as the original, 

 and into one unifacial surface with the same number of twists. 



Designating by a any positive integral number, the following table 

 of numerical results can be given: 



Number of twists and divisions. 



Original Surfaces. 



The number n is of course a positive integer. If «>o, any loup- 

 cut divides the corresponding unifacial surface into a bifacial and a 

 unifacial surface which enclose each other. Drawing the loup-cut 

 in the middle, the surface becomes a bifacial surface with a knot. 

 This knot has a certain character depending on the number « and it 

 shall be the object of my next consideration. 



The new surface has 2(2;; — i) twists of the same sense and its knot 

 is independent of these twists, so that the knot can be pushed along 

 the whole surface. Any part of the surface can be interchanged 

 with any other part; and since no part has a particular determined 

 position it must be possible to give to the surface a symmetrical shape 

 in which the twists and the knots are conspicuous. 



I shall make such an arrangement so as to have the configuration 

 in a plane. A point-to-point correspondence between a limited part 



