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rays c,c’ (cuspidal tangents), and consider the correspondence between 
the points Z,L’, which c,c’ determine on the straight line J. 
If c is made to revolve round Z, the null-points of c describe a 
curve of order (5n—8), which has a node in ZL (cf. §7). Toa point 
L therefore belong (5n—8) points C and (5n—8) points LZ’. The 
point al represents two coincidencies L = L’. The remaining coinci- 
dencies arise from cuspidal tangents w of unicuspidal points U. The 
locus of the unicuspidal points is therefore a curve of order 2(5n—9). 
This may be confirmed in the following way. If C describes the 
straight line p, the null-rays c,c’ envelop a curve of order (5n—8) 
which has p as (5n—10)-fold tangent. It therefore has, not counting 
the points of contact, (52-—8)(52—9)—(5n—10)(5n—-9), consequently 
2(5n—9) points in common with p. In each of these points the 
null-rays c and c’ have coincided. 
11. The system S@ produces in a still different way a null- 
system. Any point F is fleenodal point for five curves g”. In order 
to find this out we have only to consider the curve that arises if 
we make every d” that has F’ as node, to intersect its tangents d,d’. 
This C+? namely, has in F’ a quintuple point ’). 
I now associate to each point F as null-point the five null-rays f, 
which are inflectional tangents for the five fleenodal curves gp". 
Any point D of the straight line a is node for a dr, which 
touches the ray PD in D. I now determine the order of the locus 
of the groups of (n—3) points # which each of the curves d* has 
moreover in common with PD. If £ lies in P, 0” belongs to a 
complex S®). According to § 2 there are on a (4n-—5) nodes of 
curves 0” of (3) which send their tangent d through P. So P is 
(4n—5)-fold point of the curve (EZ) and the latter consequently of 
order (Sn—8). In each of its intersections F with a a curve ¢ has 
a fleenodal point, the inflectional tangent of which passes through P. 
From this it ensues that the locus of the null-points F of the 
rays f out of a point P (null-curve of P) is a curve of order 
(5n—8). As it must have a quintuple point in P, an arbitrary 
straight line f therefore contains (5n—13) null-points. *) 
1) In a point S (§ 8) the c” with triple point replaces three of the curves 
en; for the other two the inflectional tangent lies along one of the two fixed 
tangents d, d’. 
For a unicuspidal point (§ 9) one of the curves pr has its inflectional tangent 
along the fixed tangent d. 
2) For n=3 is on —13=2. Each @5 is then the combination of a straight 
line f and a g°. Each straight line f belongs in S(4) to a figure (f, ¢%); its 
intersections with p? are the two null-points F. 
