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A vector in the Cartesian plane being nowhere zero or infinite, of 
which the ovigin describes an are of simple curve a and the endpoint 
as a continuous function of the origin another are of simple curve 
hb, starting in P and ending in Q, deseribes as integral of its 
infinitely small variations of direction a certain total angular variation. 
If we substitute for 4 another are of simple curve 6’ starting 
likewise in P and ending in Q, the two following theorems hold: 
Lemma 1. Zf a has no point in common with b + b', and if a 
is not separated from the infinite by b+-0', then the substitution of 
b' for b causes no change in the total angular variation of the vector. 
In that case we can namely construct a closed curve c containing 
a in its outer domain, 4+ 4’ in its inner domain, and we can 
perform the modification of 6 into 6’ in a continuous way and 
entirely in the inner domain of c. The total angular variation of 
the vector ean then on one hand undergo only continuous modifi- 
cations, and on the other hand it can only vary by multiples of 
2a; thus it remains unchanged. 
Lemma 2. If b and b' form together a simple closed curve con- 
taining a in its inner domain, then by the substitution of 5' for b the 
total angular variation of the vector increases or decreases by 2x, 
according to the positive sense of circuit of the closed curve corre- 
sponding to a movement of P to Q along b' or along b. 
If namely of a vector the endpoint describes a simple closed curve 
in a positive sense, whilst its origin describes as a continuous function 
of the endpoint a closed curve lying entirely inside that simple closed 
curve, we can by means of continuous modification, which does not 
change the total angular variation of the vector, transform the curve 
described by the endpoint into a circle, and reduce the curve described 
by the origin to the centre of that circle. Thus also before this 
modification the total angular variation is equal to 22%, from which 
lemma 2 immediately ensues. 
§ 2. 
The invariant point of the circular 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 a certain circular continuum #/ passes thereby into itself. 
We represent «' together with certain surroundings yw’ uni-univa- 
lently and continuously on a region of a Cartesian plane, whereby 
