oo4 
domains determined by these curves are each submitted to a 
continuous one-one representation on w‚ and that either all with 
degree +1 or all with degree —1. By means of an indefinitely 
small modification a Canonical representation can be transformed 
into a simply ramified Riemann representation, i.e. into a represen- 
tation which in the sense of analysis situs is identical to a simply 
ramified representation of a Riemann surface with 2 sheets and of 
genus zero on the complex plane. That all simply ramified Riemann 
representations belong to the same class, follows, according to a 
remark made by Krein), out of a known theorem of Lirora—C iusscu. 
In order to transform an arbitrarily given univalent continuous 
representation « of w on w into a representation in a single point, resp. 
into a canonical representation, we first modify it continuously 
into a simplicial approximation’) a', to which we have imparted, 
by means of eventual subdivisions of the corresponding simplicial 
divisions of mu and w, the property that any base triangle of 
u covers in « either a single base triangle, or a single base side, 
or a single base point; we then investigate the possibility of finding 
two base triangles of u, one positively and the other negatively 
represented, allowing that we pass from the one to the other by 
transversing exclusively base sides of u not represented in a single 
point. If this be the case, u will possess a positively represented 
base triangle f and a negatively represented one 7, both represented 
in the same fundamental triangle / of u’, allowing us to pass from 
the one to the other by transversing exclusively such base sides of 
u, as are represented in the same side s, of 4. The base triangles 
bloet bnr of mw -erossed on this way leading trom 7. fos, ane 
then also represented entirely ins, | 
Let s, and s, be the other two sides of 4; by a continuous modi- 
fication of «/ and a suitable farther subdivision of ¢,, ¢,, .. „ln, tn 
we can generate a representation «” for which all the triangles 
titay +, H2v, tate represented. entirely -an <s, sand sand which 
possesses still the same property as «’, viz. that any base triangle 
either a single base triangle, or a single base side, 
of u covers in u’ 
or a single base point. 
In the same manner as we transformed «’ into e’, we transform 
ak 
a if possible into «'”, and we continue this process until after a 
!) Compare: “Ueber Riemann’s Theorie der algebraischen Funktionen und ihrer 
Integrale”. Leipzig, 1882. 
2) Mathem. Ann. 71 (1911), p. 102. 
