357 
and apply the operation of the first case to the pairs of fundamental 
triangles ABD and CBD; BCG and FCG; BAG and FAG 
successively *). 
By applying this operation successively to sp, sp,’—),..., sp", and 
sp’, we experience that the representation aw) of sp, on u/ can be 
transformed by a continuous modification leaving the images of the 
points P,, invariant, into a representation a, of sp, on mw’, which 
follows from a») by means of a continuous one-one correspondence 
between sp, and sp,. As sp,” can be divided into elements each 
of which is submitted for a») to a one-one representation uf degree 
+ 1 on a base triangle of u’, it is clear that sp, can be divided into 
elements each of which is submitted for a, to a one-one representation 
of degree + 1 on a base triangle of u’. The representation a, of 
sp, on wu’ is therefore a Riemann representation, and eventually it 
-may be transformed by an indefinitely small modification leaving 
the images of the points /,- invariant, into a simply ramified 
Riemann representation. 
By executing this process of modification for all the values of » for 
which it is applicable we arrive at a representation a, being for 
any of the spheres sp,, sp,,...,spr either a simply ramified, positive 
or negative Riemann representation, or a representation nowhere dense. 
In each domain g, we approximate the boundary parts y,- by 
simple closed curves x,- not intersecting each other. Each x,- includes 
with the corresponding y‚- a domain g’,-, and the z,, situated in the 
same domain g, include together a domain g',. The domains g',- be- 
longing to the same r form together a domain g’-. By means of a 
continuous series of univalent continuous representations of g, on 
sp, we can transform identity into a representation which for 
g, with the exclusion of its boundaries is a continuous one-one 
representation on sp,, whilst x,- and #',- are represented in P,-. By 
doing this for all values of » we transform a, into a representation 
a, being for each of the domains g’, and g"- after contraction of its 
rims into points either a simply ramified, positive or negative Riemann 
representation, or a representation nowhere dense. 
The domains 7, and g":, which will be represented henceforth 
by 91,92, + + 5%, are determined on u by a finite number of simple 
closed curves not intersecting each other. | 
1) If we dropped the condition of the invariancy of the images of the points 
P‚- (introduced only for the sake of clearness), this second case might have been 
treated of course in the same manner as the first. 
