( 727 ) 
the sector by a finite or denumerable number of curves of the second 
kind, not enclosing each other, and not 
approaching P indefinitely. These do¬ 
mains we take from the sector (conse¬ 
quently modify an arc of c), and there 
remains a new sector, bounded by the 
same base curves as the old one, but 
consisting of one leaf inside which lie 
only curves of the first kind. This leaf 
we can fill with curves of the first kind 
not crossing each other (see fig. 4). 
These sectors of the second case, 
which are reduced to a single leaf, Fl S* 4 Elliptic sector, 
we shall call elliptic sectors. 
We now pass to the discussion of a sector of the second category, 
of which, to fix our ideas, we assume, that it is bounded by two 
positive base curves. 
Let us consider the set of points lying in the sector or on its 
boundary, through which curves of the second kind not crossing 
the base curves can be drawn. This set of points cannot approach 
P indefinitely, as otherwise it would give rise to a negative curve 
of the third kind not crossing the base curves, which is excluded. 
In the same way as for the elliptic sectors we destroy the regions 
covered by this set of points, and there remains a sector of the 
second category bounded by a modified arc of c , inside which no 
curves of the second kind not crossing the base curves can be drawn. 
In the modified sector we now consider the set of points, through 
which curves of the first kind not crossing the base curves can be 
drawn, and it is clear that this set of points cannot indefinitely 
approach the just now modified curve c. The regions covered by it 
are therefore bounded by a finite or denumerable number of curves 
of the first kind, not enclosing each other, 
not indefinitely approaching c, and each 
enclosing a domain which forms a leaf, 
not differing from those appearing in 
the hyperbolic sectors. 
By the method applied above already 
several times the region outside the 
leaves can be filled with curves of the 
third kind (for instance we can choose 
Fiz 5 Parabolic sector. for them the system of base curves 
49 
Proceedings Royal Acad. Amsterdam. Vol. XII. 
