358 
We choose an arbitrary domain g,, and suppose in the first place 
that « is for the sphere o, into which a, is transformed by contrac- 
tion of its rims into points, a simply ramified Riemann representation. 
We then draw on wa system of ramification sections belonging to 
this representation and corresponding to a system of simple closed 
“ramification curves” on 5, By first leaving the ramification sections 
on u’ invariant and varying eventually continuously the ramification 
curves on 6, in such a manner that after that they contain no more a 
point corresponding to a rim of %,, and then leaving the ramification 
curves on 6, invariant and contracting the ramification sections 
on u continuously into points, we can transform the representation 
of 6,on w determined by «/ continuously into a canonical representation. 
During this continuous modification the points representing the rims 
of g, vary also in general. Let 7, be such a rim and 4,- the residual 
domain of 9, on u determined by /. We then can follow the con- 
tinuous variation of the image point of / by a continuous series of 
continuous one-one transformations of win itself to which corresponds 
a continuous modification of the representation of g,- on w determined 
by a. By applying this modification to the representations of all the 
residual domains of 9, we generate a representation @; of u on uw 
into which «, can be transformed continuously, and which is a 
canonical representation for 6,. 
In the second place we suppose «a, to be for 6, a representation 
nowhere dense. Then we can modify the representation of 6 on u 
determined by «; into a representation in a single point. The varia- 
tion of the image points of the rims of 3, implied by this modifica- 
tion, can be followed once more in the way deseribed above by a 
continuous modification of the representation of the residual domains 
of ¢,, furnishing us with a representation @; of u on u into which 
« can be transformed continuousiy, and which represents 6, in a 
single point. 
By executing this operation for all values of » successively, we 
get a representation Ee of u on w, into which «a can be trans- 
formed continuously, and which represents each of the domains 
§,, Jot either after contraction of the rims into points canoni- 
cally, or in a single point. The sphere u is now divided by a finite 
number of non intersecting simple closed curves into a finite number 
(5) 
l 
domains is submitted either after contraction of the rims into points 
to a continuous one-one representation, or to a representation in a 
single point. Thus the degree of these representations is 0, +1, or 
of domains d,, d,, .. . ., dy in such a way that for @” each of these 
