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happen only once that it undergoes no division; after that namely 
it becomes singly connected, so is divided at each new insertion of 
a tangent curve (see fig. 6). 
We finally add the limit curves, and we prove in the same way 
as in § 2 that then through each point of the inner domain of c 
passes a tangent curve. 
A point zero being in the second principal case we shall call a 
rotation point 
So we can say: 
Theorem 4. An isolated singular point is either a rotation point, 
or a vicinity of it can be divided into a finite number of hyperbolic, 
elliptic, and parabolic sectors. 
The filling of a vicinity of a non-singular point in $'2 furnishes 
in this terminology two hyperbolic and two parabolic sectors. 
We must add the observation that in the most general case, where 
neither in a singular, nor in a non-singular point the tangent curve 
is determined, sometimes by a modified method of construction, the 
structure of the first principal case can be given to a vicinity of a 
point zero being in the second principal case. 
Even the form of the sector division of the first principal case is then 
not necessarily unequivocally determined. Out of the reasonings of the 
following § we can, however, deduce that, if modifications are 
possible in the form of the sector division, the difference of the number 
of elliptic sectors and the number of hyperbolic sectors always 
remains the same. 
§ 5. 
The reduction of an isolated singular point. 
For what follows it is desirable to represent the domain y on a 
Euclidean plane, and farther to substitute for the curve c a simple 
closed curve c' emerging nowhere from c, containing likewise P in 
its inner domain, and consisting of arcs of tangent curves and of 
orthogonal curves. In the second principal case this is already 
attained, and in the first principal case we have to modify in a 
suitable way only those arcs of c which bound the hyperbolic and 
the parabolic sectors. 
In a hyperbolic sector we effect this by choosing a point on each 
of the two bounding base curves, and by drawing from those points 
H and K inside into the sector orthogonal arcs not intersecting one 
another. Then there is certainly an arc of x a curve of the second kind 
49* 
