( 295 ) 
Furthermore we have supposed in our reasoning, that K, otc. 
have nothing more in common with T, than Z,. etc. However it 
might be possible, that the last part of e. g. the branch of K r ap¬ 
proximating Z, is contained in One can easily be convinced, 
that this pecularity does not harm the proof. 
We can sum up our results as follows: 
Theorem 1. An arbitrary continuous one-one transformation of a 
twosided surface in itself with invariant indicatrk possesses a trans¬ 
formation domain, which either breaks the connection of the surface , 
or joins two isolated open systems of cui'ves , invariant for the trans¬ 
formation. 
As furthermore such an invariant open system of curves possesses 
at least one invariant point, as I shall prove in another communi¬ 
cation, the following holds likewise: - 
Theorem 2. An arbitrary continuous one-one transformation of a 
twosided surface in itself with invariant indicatrix possesses a trans¬ 
formation domain, which either breaks the connection of the surface, 
or joins two points invariant for the transformation. 
The fundamental importance of these theorems for the theory of 
transformations and transformation groups I shall show lateron. 
Here I only wish to indicate, how the theorem of the invariant 
point of the sphere, proved in my first communication upon this 
subject, is contained in them, and how this theorem can be extended 
to the Cartesian plane. 
For, if the transformation domain of a sphere or Cartesian plane 
has more than two rest regions, then surely one of them must be 
invariant together with all the points of its boundary. 
If we thus exclude this case, a transformation domain 0 on the 
sphere, which breaks its connection, is either annular, or singly con¬ 
nected ; fig. 9 shows either of these possibilities. 
Fig. 9. 
