163 
As the triplets of tangents of the curves rt” of that pencil form 
an involution, S is triple point with a cuspidal branch for four 
curves +”. Each singular null-point, therefore, bears four double 
null rays. 
18. The null-curves of P and Q have the singular null-points S 
and the null-points of PQ in common. Each of the remaining 
intersections 7’ sends a null-ray through P, a second through Q. 
From (7n—15)?—21(n—2)’—(7n—18) it therefore ensues that the 
null-rays ft, will envelop a curve of class (28n’—133n+159), if 
t, revolves round a point P. The null-rays of P belong each twice 
to this envelope, each of the remaining tangents, which it sends 
through P, is evidently double null-ray. The double null-rays, therefore, 
envelop a curve of the class (28n?—133n-+158). 
19. In order to find the locus of the points 7’ for which two 
of the null-rays coincide, 1 shall consider the curve (p)7z,—15 enve- 
loped by the null-rays of the points lying on p. It has p as (?7n—18)- 
fold tangent, is therefore intersected by p in (7n—15)(7n—16)— 
(7n—18)(7n—17) points. As for each of these points two null-rays 
coincide, the points 7’ with double null-rays lie on a curve of order 
(28n—66). 
It is at the same time the locus of the triple points that have a 
cuspidal branch. 
For n= 3 we have a null-system (3,3); the curves t° are three- 
rays in that case. An arbitrary straight line then forms figures c* 
with the curves of a net of conics. The Jacobian of that net deter- 
mines the three null-points of the straight line. 
If the system S‘) has three base-points, the three null-points of 
a straight line are produced by the intersection of the sides of a 
triangle, which has the base-points as vertices. Each base-point is 
the centrum of a pencil of singular null-rays. 
