55 
Let P be an arbitrary point coinciding neither with P?, nor with 
P, and passing by ar into AR, and let Q be the point corresponding 
in latitude with P and in longitude with . Then by transforming the 
different points / continuously and uniformly along circles of longitude 
into the corresponding points Q we define a continuous series o of 
univalent continuous transformations of the sphere in itself with the 
property that of none of the points & the path passes through P, or P,. 
So an arbitrary curve c/ is transformed by @ into a curve c", with 
respect to which P, possesses likewise the order + 1, so that c" 
covers the corresponding circle of latitude e with the degree’) + 1. 
From this ensues that an are of a circle of latitude connecting 
an arbitrary point P with the corresponding point Q defines une- 
quivocally for any point P an art of circle of latitude PQ whose 
variation with P is uniformly continuous, so that it is possible to 
construct a continuous series 9’ of univalent continuous transformations 
of the sphere in itself, transforming each point Q into the corre- 
sponding point P, and thereby the transformation ate into identity. 
But then tee’ is the looked out for continuous series of transfor- 
mations, transforming « into identity. 
We shall say that two transformations belong to the same class, 
if they can be transformed continuously into each other. We then 
can state the theorem proved just now in the following form: 
Trrorem 1. All continuous one-one transformations with imvariant 
indicatrix of a sphere in itself belony to the same class. 
As the continuous one-one transformations with invariant indicatrix 
form a special case of the univalent continuous transformations of 
degree -+ 17), the question arises whether perhaps theorem 1 is a 
special case of the more general property that all the univalent 
continuous transformations of the same degree of a sphere in itself 
belong to the same class. We shall see that this is indeed the case; 
we shall namely show that any univalent continuous representation 
of degree zero of a sphere u on a sphere u’ can be transformed by 
continuous modification into a representation of u in a single point 
of w, and that- any univalent continuous representation of degree 
nz0 of a sphere u on a sphere u’ can be transformed by continuous 
modification into a canonical representation of degree n, i.e. into a 
representation for which n—1 non intersecting simple closed curves 
of u are each represented in a single point of u, whilst the n 
1) Mathem. Ann. 71 (1911), p. 106. 
*) Mathem. Ann. 71 (1911), p. 106 and 324. 
