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6. If the complex {c*} has a base-point B, it is at the same time 
singular null-point, for two points on a ray passing through B, 
determine a nodal d°, with node in 5. The double rays of the 
involution formed by the curves d? with node B are double lines 
of {c*}, consequently singular null-rays. Other double null-rays do not 
exist, for if a straight line d of d? does not pass through B, d’ does. 
As B is node of the Jacobian of each net belonging to {c’), this 
point replaces four critical points. Two more singular points, there- 
fore, lie outside B; they are connected by a singular null-ray. 
7. In a fourfold linear system S, each point D is node to a 
pencil (d”). Two of those curves have a cusp in C= D. 
I now consider the nz/-system in which to the null-point C are 
associated the cuspidal tangents c,c’ of the two cuspidal curves yr, 
which have their cusps in C. 
The straight line d is touched in each of its points D by a nodal 
de, which has its node in D. With the straight line PD dr” has 
moreover (n—2) points / in common. In order to find the locus of 
the points /, I shall inquire how often / gets into P. In this case 
J” belongs to the complex that has a base point in P; in it occur 
(4n—7) d”, which touch at d ($ 2). Consequently (#) is a curve of 
order (52—9). 
If # lies on d, PE =d’ touches in that point at a òr, which 
has its node on J. Every straight line d therefore is nodal tangent 
of (5n—9) curves 6”, of which the second tangent d’ passes through 
P. If dis now made to revolve round a point Q, the point D. 
describes a curve (D) of which every point is node of a 0”, which 
sends its tangents d and d’ through Q and P. In Q a dis touched 
by QP, so Q and consequently P is a point of (D), so that this 
curve is of order (5n—8). 
If C is one of the (5n—10) points, which (D’) has in common 
with the straight line PQ, besides P and Q, the tangents d,d’ fall 
both along PQ, so that C is a cusp of a cuspidal curve y", which 
has c= PQ as cuspidal tangent. 
In the above null-system a straight line therefore has 5(n —2)null- 
points. 
If e revolves round a point M, the null-points C describe a curve 
of order (5x—8), with node M (the null curve of MM). 
8. The system S@® contains a number of curves with a triple 
point. If S“ is represented by the equation 
A+ PB 4 yC + dD+ el =0, 
