( 856 ) 
furnishes a single closed tangent curve 7, with the same property 
as above. 
y. Neither when we pursue, neither when we recure it meets 7,. 
It then converges on both sides to a single closed tangent curve. 
One of these can coincide with 7,; the other however is a single 
closed tangent curve 7, with the same property as above. 
d. When we pursue as well as when we recure it meets 7,. The 
are lying between the first meeting-points on both sides forms then 
with an arc of 7, joining those same points a single closed tangent 
curve 7, with the same property as above. 
In the same way we can now again enclose a part of the inner 
domain of r, by a single closed tangent curve 7,, and we can 
construct in this way a fundamental series of single closed tangent 
curves 
Ta Vas ila lias Mien See 
for which we show, in the same way as above for the curves £, 
that there exists an upper limit for their length of are and 
fartheron, that they converge uniformly to a single closed tangent 
curve 7, whose inner domain is a part of that of any curve 7. 
But we can still again let 7, lose a part of its inner domain by 
a single closed tangent curve r,4:, and again 7,41 by rage, and 
this process can be continued after any index of the second class 
of numbers. 
On the other hand, however, this is an absurdity, as the system 
of those losses of domain must remain denumerable. 
The supposition, that the vector direction should be determined in 
any. point, has thus proved to be impossible, so that we can formulate : 
Trrorem 1. A vector direction varying continuously on a singly 
connected, twosided, closed surface must be indeterminate in at least 
one point. 
And from this follows directly : 
TuroreM 2. A vector distribution anywhere unwalent and continuous 
on a singly connected, twosided, closed surface must be zero or 
infinite in at least one point. 
If we represent the complex plane stereographically on the NEUMANN 
sphere, a complex function becomes a vector distribution on the sphere. 
So we can also interpret our result as follows: 
THrorEM 3. A wnivalent, continuous function of a complex variable 
