( 797 ) 
point, the row of signs must precisely be pd. Thus the assumption, that 
from a, up to and inclusive of ap no invariant point appears, leads 
to an inconsistency, and we have proved: 
Turorrm 1. A continuous one-one transformation in itself with 
mvariant indicatrie of a singly connected, twosided, closed surface 
possesses at least one invariant point. 
-The above given proof fails for a transformation with inversion 
of the indicatrix; on the contrary, according to that proof we can 
show how such transformations may easily be constructed without 
invariant point. 
For then the curves 4, and k#', have the same sense of circuit, 
and with the aid of a method of representation of ScHOENFLIES *) 
we can arrange them to possess for all values « between a, and 
a, only two points of intersection whilst originally &,, lies inside 
k',,. Let us now regard the part of the boundary of y, belonging 
to kz, then fig. 7 shows how the image of that arc can begin to 
de ij SS 
wa 
Ane eee 
4 \F = Se 
a “5 
/ Je Ds 
a, \ 
, 4 
te: 
a 
: \ 
\ 
Se 
ve 
\ 
| ee \ Ky 
| ! ‘ 
| 5 
\ | 
\ 
Ps { » 
\ 
| RIEN 
5 / 
ask / 
Fig. 7. 
1) Mathem, Ann. 62, p. 319—324. 
| 54 
Proceedings Royal Acad. Amsterdam. Vol. XI. 
