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each other, then there are among them which cut from the sector 
an area as small as one likes, and at the same time the maximum 
distance, which such a series reaches from c, decreases under each 
finite limit. 
And analogously, if we draw in the sector a well-ordered series, 
continued as far as possible, of curves of the first kind enclosing 
each other, it converges either to a curve of the first kind, or to two 
curves of the third kind and between them a finite or denumerable 
set of curves of the second kind, not enclosing each other, and not 
approaching P indefinitely. 
If we can construct an infinite number of such series not enclosing 
each other, then there are among them which enclose an area as 
small as one likes, and at the same time the maximum distance, 
which such a series reaches from P, decreases under each finite limit. 
From this ensues that for the sectors of the first category we have 
to distinguish two cases: 
First case. There are curves of the second kind in indefinite 
vicinity of P. Then the domain of the curves of the second kind is 
bounded by the two base curves which bound the sector, and a 
finite or denumerable number of curves of the first kind, not enclosing 
each other, and not approaching c indefinitely, in whose inner domains, 
which we call the leaves of the sector, can lie only curves of the 
first kind. 
The region outside the leaves can be covered as follows with, curves 
of the second kind not crossing each other: we first construct one 
which reaches a distance as great as possible from c and the boundary 
of the leaves; in this way two new regions are determined, in each 
of which we repeat this insertion. This process we continue indefini¬ 
tely, and finally we add the limit curves. That then the region 
outside the leaves is entirely covered, 
is evident from the reasoning fol¬ 
lowed in § 2. 
And in the same way we fill each 
of the leaves with curves of the first 
kind not crossing each other. The 
whole of the tangent curves filling 
the sector finally gets the form in- 
dicated in fig. 3. The sectors being 
in the discussed first case we shall 
Fig. 3. Hyperbolic sector. call hyperbolic sectors. 
Second case. There are no curves of the second kind in indefinite 
vicinity of P. Then the domains covered by these curves are cut off from 
