( ffi ) 
If the pursuing branch of r lies in the outer 'domain of u x , the 
preceding holds with slight modifications. A point of the limit set 
of the Wfc’s now' necessarily bounds a region belonging to y, and 
lying outside all Uk s, only then when it is not a point of the 
boundary of y. The inner circumference, to which r now converges 
spirally on the inner side, consists here again of arcs of simple curve, 
which are tangent carves to the vector distribution, but these tangent 
curves can lie entirely Or partially in the boundary of y. 
However they have all again a pursuing sense belonging to the 
same sense of circuit Of the circumference. 
We now agree about the following : When a pursuing branch of a 
tangent curve reaches a point zero, we continue! it, if possible, along 
a pursuing branch, starting from , that point zero, and not meeting 
the former within a certain finite distance; but if such a continuation 
is impossible, we stop the branch at that point zero, and so we do 
likewise when the branch has entered into a closed curve or has 
approximated spirally a circumference. Then we can. resume the 
preceding reasonings as follows: 
Theorem 2. A tangent curve is either a simple closed curve, or 
save its ends it is an arc of simple curve, of which the pursuing as 
ivell as the recurring branch shows one of the following characters-. 
1 st . stopping at a point of the boundary _of y, 2 nd . stopping at a, 
point zero; 3 rd . entering into a simple closed tangent curve; 4 th . spirally 
converging to a circumference , consisting of one or more simple 
closed tangent curves. 
From this ensues in particular: 
Theorem 3. A tangent curve cannot return into indefinite jncmity 
of one of its points, after having reached a finite distance from it, 
unless it be to close itself in that point. 
That the Tast’theorem is not a matter of course, is evident from 
the fact that it does not hold for an annular surface. On this it is 
easy to construct tangent curves of the form pointed out by Lorentz 
(Enz. der Math. Wiss. Y 2, p. 120,121). 
We finally notice that the vector distribution considered in this §, 
does not possess of necessity a singular point (as is the case on the 
sphere). This is proved directly, by considering in the inner domain 
of a circle, situated in a Euclidean plane, a vector every where constant. 
§ 2 . 
The structure of the field in the vicinity of a rum-singular point. 
To classify the singular points we shaft surround each of them 
