( 855 ) 
So, as any point of & is limit point of points of A” lying on7, 
the limit curve hk is a tangent curve to the vector distribution. ; 
Farthermore s also represents the length of arc of 4) and as the 
lengths of the &™’s remain below a same finite limit, ®) also 
returns into itself after having described a finite length of are. 
The curve 4@) cannot reduce to a single point, for then the 
whole of the directions of the tangents to a curve contracting to a 
single point would converge to a single direction, namely the vector 
direction in that limit point, which is impossible. 
Neither can a point of 4@ belong to two different values of s, 
unless after a whole circuit; for otherwise 4) would consist of a 
single closed curve plus points in its “inner domain” (if we call 
its “outer domain” that in which all curves 4 lie); which is 
likewise impossible. 
So it is evident that © is a single closed curve to which the 
pursuing branch of 7 spirally converges uniformly. 
In the same way it is evident, that also the recurrent branch of r 
spirally converges uniformly to a single closed curve 4’) lying 
entirely outside 4%). 
The tangent curve r has therefore for analysis situs the character 
of a double circular spiral cireuit, whose two asymptotic closed 
curves are likewise tangent curves to the vector distribution. 
So we possess for continuous vector distributions having in any point 
a definite direction, at any rate a single closed tangent curve. 
Possessing now such a closed tangent curve 7,, we can start 
another tangent curve in one of the domains determined by the former, 
which domain we shall call its “inner domain”. This curve ean 
behave in different ways: 
A. When sufficiently pursued on one side and recurred on the 
other side, it finally returns into itself without having met DS 
In this case we possess a single closed tangent curve r, bounding 
a singly connected “inner domain” forming a part of the inner 
domain of 7,. 
B. It does not return into itself inside 7,; here the following cases 
are possible : 
a. When recurring we find it starting somewhere on 7, ; when 
we pursue it, if does not meet 7,. Then according to what precedes 
it the pursuing branch converges spirally to a single closed tangent 
curve 7,, bounding a singly connected “inner domain”, which is, a 
part of the inner domain of 7,. 
8. When recurring we do not find it starting on 7, ; when we pursue 
it however- it ends somewhere on r,. Then the recurrent branch 
