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Mathematics. — “On continuous vector distributions on surfaces”: 
(2 nd communication) 1 ). By Dr. L. E. J. Brouwer. (Commu¬ 
nicated by Prof. D. J. Kobteweg). 
(Communicated in the meeting of February 26, 1910). 
f 1- 
The tangent curves to a finite, uniformly continuous vector distri¬ 
bution with a finite a ) number of singular points in a singly connected 
inner domain of a closed curve. 
Let y be the domain under consideration, then we can represent it on 
a sphere, so we can immediately formulate on account of the property 
deduced in the first communication (see there page 855): 
Theorem 1. A tangent curve, which does not indefinitely approach 
a point zero, is either a simple closed curve, or its pursuing as well 
as its recurring branch shows one of the following characters: 1 st . 
stopping at a point of the boundary of y ; 2 nd . spirally converging 
to a simple closed tangent curve-, 3 rd . entering into a simple closed 
tangent curve . 
We now shall farther investigate the form (in the sense of analysis 
situs) of a tangent curve r, of which we -assume, that at least on*? of 
the two branches (e. g. the pursuing branch) approaches indefinitely 
one or more points zero, i. e. singular points of the vector distribution. 
We start the tangent curve in a point A„ (not a point zero) and we 
pursue that curve in the following way : By we understand a 
distance with the property that in two points lying inside the same 
geodetic circle described with a radius fts, and possessing both 
a distance e from the points zero, the vectors certainly make an 
fl ngle <C g ^ with each other. We farther choose a fundamental series 
of decreasing quantities e u -converging to 0, and of corre¬ 
sponding decreasing distances --- which all we suppose, 
if a is the distance of A 0 from the points zero, to be smaller 
than a—e 1# 
We then prove in the manner indicated in the first communication 
P- »52, that, when pursuing r from 4 0 , a point B 0 is reached, 
possessing a distance from A a ; we call the arc A 0 B 0 a & r arc. 
According to our supposition there now exists a finite number n x in 
For the first communication see these Proceedings Vol. XT 2 -p> 850. 
*) This restriction we shall drop in a Mowing cbmmunication.’ 
