The invariant point of the parabolic continuum. 
We suppose a two-sided surface to be submitted with invariant 
indicatrix to a continuous one-one transformation in itself in such a 
way that by it a certain nowhere dense parabolic continuum gp’ is 
transformed into itself. 
We suppose that g’ possesses no point invariant for the trans- 
formation. Then its circumference cannot contain an invariant Schnitt 
either; for, this would be a circular continuum and therefore according 
to theorem 1 would give rise to an invariant point. 
We represent gy’ together with certain surroundings wp’ uni-uni- 
valently and continuously on a region of a Cartesian plane whereby 
they become respectively the images p and w. All the figures of the 
Cartesian plane to be constructed in the following and likewise their 
images for the transformation and their “counterimages” (i.e. their 
images for the inverse transformation) we suppose to lie in w. We 
suppose fartheron that for a positive circuit of the circumference 
of p each accessible point precedes its image. 
We surround p by a fundamental series of polygonal lines ,, },, },,... 
lying inside each other and approximating p at indefinitely decreasing 
distances €,,&,&,,---, and we draw to an accessible point A 
lying on the circumference of g a path w cutting each B, once and 
not more than once, and being < de, *). 
By a circumference domain of p we understand such a segment 
of the linear type of order of its accessible points, as lies entirely 
outside its image segment, but for which each extension causes that 
property to be lost. 
Let Y and } be two accessible points on the circumference of p 
which are separated by A, whilst the order of succession YAY 
corresponds to a positive sense of circuit and between X and A as 
well as between A and Y there exist at least three circumference 
domains lying outside each other. 
Let B be an accessible point of p preceding X for a positive 
sense of circuit, possessing a finite distance p from the circumference 
segment YJ’, and belonging to the boundary of both regions deter- 
mined by w between p and },. 
Let us understand by U an arbitrary accessible point of the 
1) Compare Scroexrues, Bericht über die Mengenlehre II, p. 127; L. E. J. Brouwer, 
“Zur Analysis Situs’, Mathem. Annalen, Vol. 68, p. 428. 
