(EER 
So we have in this second case, defining the angle y; in the same 
manner as in the first case: 
Bk = yr + 2m. 
3rd, On the circumference of p the segment Ok Qi is a part of 
the segment Ry Pr (see fig. 3). Then the image skeleton arc 
Qk Tk Ald er lies entirely inside W, and we choose as path are 
Tt the image skeleton are tur itself. Then we have in this 
third ease, defining the angle y; in the same way as in the first 
and second cases: 
A Yk: 
Now we can take Sy; as the total angular variation of a vector 
nowhere becoming zero, of which the origin describes Din « positive 
sense and the endpoint as a continuous function of the origin a 
closed curve passing nowhere outside 9, so that we have 
D= df 
From this ensues in connection with the preceding formulae: 
Soe — on, 
where n represents a positive integer > 1. 
Hence 9 cannot be equal to zero, from which we conclude that 
the distribution of the transformation vectors must possess inside 
at least one singular point, i.0.w. that, contrary to the supposition 
at the commencement of this $, there must lie inside } at least one 
point invariant for the transformation. 
With this we have proved: 
Tnrorem 1. For a continuous one-one transformation with invariant 
indicatrix of a two-sided surface in itself an invariant circular con- 
tinuum contains at least one invariant point’). 
1) Compare Mathem. Annalen, Vol. 69, p. 178; these Proceedings Vol. XII, p. 295. 
