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insertion, and we continue this process as often as possible, evenr 
tuallj to an indefinite number of insertions. 
If in this manner we have obtained an infinite number of tangent 
curves of the third kind, they determine limit elements which each 
are either again a tangent curve of the third kind, or contain such 
a curve as a part. And in particular a fundamental series of positive 
respectively negative curves of the third kind determines in its limit 
elements again positive respectively negative curves of the third kind. 
After addition of these limit curves of the third kind we are, 
however, quite sure that no new curves of the third kind not crossing 
the existing ones can be inserted. This is evident from a reasoning 
analogous to that followed in $ 2. The whole of the curves of the 
third kind, obtained now, we shall call a system of base curves of 
the vicinity of P. 
An arbitrary positive base curve and an arbitrary negative one 
enclose inside c a sector, of which the area cannot fall below a 
certain finite limit. For otherwise we should have a fundamental series 
of positive base curves, and a fundamental series of negative ones, 
possessing the same base curve as a limit element, which is impossible, 
as that limit base curve would have to be positive as well as negative. 
So the inner domain of c is divided into a finite number of sectors 
which can be brought under the two following categories: 
First category. Sectors bounded by a positive and a negative base 
curve, between which lie no further base curves. The areas of these 
sectors surpass a certain finite limit. 
Second category. Sectors bounded by two positive (respectively two 
negative) base curves and containing only positive (respectively negative) 
base curves. A sector of this category can reduce itself in special 
cases to a single base curve. 
We shall first treat a sector of the first category and to that end 
we first notice that outside a curve of the second kind lying in it 
(i. e. between that curve and c) lie only curves of the second kind, 
and inside a curve of the first kind lying in it only curves of the 
first kind. 
If we draw in the sector a well-ordered series, continued as far 
as possible, of curves of the second kind enclosing each other, then 
it converges either to a curve of the second kind, or to two curves 
of the third kind and between them a finite or denumerable set 
of curves of the first kind, not enclosing each other, and not 
approaching c indefinitely. 
If we can construct an infinite number of such series not enclosing 
