154 KANSAS UNIVERSITY QUARTERLV. 



the variables are represented. Thus a ribbon witli an even number 

 of twists would be as effective as a cylinder, and yet could not be 

 physically deformed into a cylinder or a plane. The necessary and 

 sufficient condition for the equivalence of two bifacial surfaces is, 

 therefore, that they must have the same connectivity. 



Now we can give an example which is an exception to this case. 

 The unifacial surface can be neither deformed nor transformed into 

 a bifacial surface. On two such surfaces a point-to-point transforma- 

 tion is no longer possible, and thus unifacial surfaces must be excluded 

 from the representation of a complex variable. Unifacial and bifa- 

 cial surfaces are both of connectivity two, but are altogether different 

 in character as far as the representation of a complex variable upon 

 them is concerned. From this point of view it is interesting to study 

 these surfaces in particular and in a more geometrical way. 



In Topology the problem of unilateral and bilateral surfaces, as 

 Moebius calls them, has already been treated several times, and many 

 years ago. I refer the reader who wishes to know more about the 

 investigations in Topology to the following principal authors: 



Listing: — Vorstudien zur Topologie, Goettinger Studien, 1847. 



Tait: — On Knots. Transactions of the Royal Society of Edinburgh 

 of 1879; and Kirkman in the same volume. 



Simony: — Neue Tatsachen aus dem Gebiete der Topologie, Math. 

 Annalen, Vol. XIX and XXIV. 



In these treatises may be found almost all tlie topological proper- 

 ties of unifacial and bifacial surfaces. Only surfaces come into 

 consideration which are liable to topological operations and no 

 reference is made to their employment in other investigations. 

 Starting from the theory of connected surfaces I shall make use of its 

 terminology; and I shall not only consider the result of one loup-cut 

 extended over the whole surface, but also the case of any even or odd 

 number of loup-cuts dividing or transforming the surface into other 

 surfaces of the same connectivity. 



In order to represent unifacial and bifacial surfaces in a practical 

 way, we can give a ribbon, previous to being closed, any number of 

 twists. The ribbon represents a unifacial surface if it has an odd 

 number of twists and a bifacial surface of it has an even number 

 of twists. In the first case it is always possible to get from 

 one point of the surface to its opposite point — two such points must 

 be considered as opposite points on the two faces of the ribbon — 

 without crossing the boundary; while it is not possible in the second 

 case. The origin of the words unifacial and bifacial is to be found 

 in this fact. 



Let us now draw a cross-cut in a unifacial surface perpendicular to 



