(ATL) 
We now come to the total angular variation of the transformation 
vector during the description of the are a2,0,K,P,; on the ground 
of $ 1 it ean also be obtained by earrying first the origin of the 
vector along the are a,4,A4,P, and the endpoint as a continuous 
function of the origin along the are P,R,R,P,K,L,2%,; and then, 
whilst the origin remains in Z,, carrying the endpoint still along the 
AUC TON Ds 
Finally we can obtain the total angular variation of the trans- 
formation vector during the description of the are P,R‚R,P, by 
carrying first the endpoint of the vector along p,r,9,7, whilst the 
origin remains in P,, and then the origin along P,R,R,P,, whilst 
the endpoint remains in ar. 
So the total angular variation of the transformation vector for a 
positive circuit of © is obtained by carrying first the origin of a 
nowhere vanishing vector along the are P,A,L,a7,H,H,», and the 
endpoint as a continuous function of the origin along a certain curve 
nowbere passing outside €; then, whilst the origin runs along the 
are x,L,K,P,, carrying the endpoint along the arc P,R,R,P,K,L,2, ; 
and finally, whilst the endpoint remains in 2,, carrying the origin 
along the are P,R,R,?P,. 
In none of the three parts of this movement the endpoint of the 
vector has passed outside €, so that the total angular variation 
amounts to + 27, from which we conclude that, contrary to the 
supposition, the distribution of the transformation vectors must possess 
inside © at least one singular point. 
With this we have proved: 
Trrorem 2. Kor a continuous one-one transformation with invariant 
indicatria of a two-sided surface in itself an invariant nowhere 
dense parabohe continuum contains at least one invariant point. *) 
BRR A A 
In the 28¢ communication on this subject, these Proceedings Vol. XII 
p. 289, in the note 
for: Mathem. Annalen, Bd. 68. 
read: Mathem. Annalen, Bd. 68, 69. 
p. 297, 1. 10—13 from top 
for: furthermore, if no invariant point exists, we can arrange, 
that the just-mentioned series of images of w, continued 
indefinitely on both sides, covers the whole Cartesian 
plane, i.o. w. we have proved: 
read: i. 0. w. we have proved: 
1) Compare Mathem. Annalen, Vol. 69, p. 178. 
