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have its endpoints on the boundary of y,, but how it can lateron 
withdraw its endpoints from that boundary and fartheron remain 
entirely outside it, without an invariant point having to appear. 
We thus formulate : 
Tunorem 2. A continuous one-one transformation in itself with 
inversion of the indicatrie of a singly connected, twosided, closed 
surface does not necessarily leave a point invariant *). 
An elementary special case of theorems 1 and 2 is furnished 
by a sphere in ordinary space, having always two invariant points 
for congruent transformations in itself, but not necessarily one for 
symmetric transformations in itself. 
In the formulation of theorem 1 the restriction implied in the word 
closed is not superfluous; for, an ordinary Cartesian plane has in an 
arbitrary translation a continuous one-one transformation in itself 
with invariant indicatrix, without an invariant point. 
Neither is superfluous the condition of single connection; for 
the ordinary tore of Euclidean space has in an arbitrary rotation 
about its axis a continuous one-one transformation in itself with 
invariant indicatrix without an invariant point. 
The restriction of twosidedness, however, can be cancelled. 
We can, namely, bring the points of a onesided, singly connected, 
closed surface into a continuous one-two correspondence with those 
of a twosided one; to an indieatrix on the onesided surface then 
correspond two opposite indicatrices on the twosided one. On the 
ground of such a correspondence answer to a continuous one-one 
transformation in itself of the onesided surface two such transfor- 
mations in itself of the twosided one, so that for one of them the 
indicatrix remains invariant, whilst it is inverted for the other one. 
As for the former at least one point remains invariant, for both 
at least one pair of points and for the transformation of the 
onesided surface at least one point remains invariant. 
As farthermore for that transformation of that onesided surface 
we cannot speak of remaining invariant or inversion of the indica- 
trix, we can formulate: 
Trrorem 38. A continuous one-one transformation in itself of a 
singly connected, onesided, closed surface leaves at least one point 
envariant. 
An elementary special case of theorem 3 we find in an arbitrary 
plane, real-projective transformation having at least one invariant point. 
1) For a closed line of theorems 1 and 2 exactly the contrary holds: here a 
transformation with invariant indicatrix can exist without an invariant point; but 
with inverted indicatrix such a transformation is impossible. 
