Mathematics. — “Null systems determined by linear systems of plane 
alyebraic curves’. By Prof. Jan pe Vruxs. 
(Communicated in the meeting of January 25, 1919). 
1. A triply infinite system (complex) S‘* of plane algebraic curves c” 
contains a twofold infinity of nodal curves; for an arbitrarily chosen 
point D is node of a nodal curve d” belonging to S®). 
I shall now consider the null system in which the tangents d, d' 
of dr are associated as null rays with D as null point. 
The nodal curves of a net belonging to S‘) have their nodes on 
the Jacobian, which is a curve of order 3(2—1). It has in common 
with the Jacobian of a second net the 3(m—1)* nodes, which occur 
in the pencil common to the two nets. The remaining intersections 
of the two loci are critical points, i.e. nodes for the curves of a 
pencil. The null system, therefore, has 6(n—1)’ singular null points. 
2. Let a be an arbitrary straight line, P an arbitrary point. The 
dr, which has its node D on a, intersects the ray PD, moreover, 
in (n—2) points #. If Eis to get into P, dr must belong to the 
net that possesses a base-point in P; D lies then on the Jacobian 
of that net. The locus (£) of the points / passes, therefore, 3 (n—1) 
times through P, and is consequently a curve of order (4n—5S). 
Each intersection of (#) with a is node of a 0”, of which one of 
the tangents d passes through P. 
There is therefore a curve (D)p of order (42—5) which contains 
the nodes of the nodal curves 9”, which send one of their tangents 
d through a given point P. It will be called the null-curve of P. 
For a singular point S it has in S a triple point. As P evidently 
is node of (D), there lie on a ray d passing through P (4n—7) points 
D, for which d is one of the tangents of the corresponding curve 
dr. From which ensues: an arbitrary straight line d has (4n—7) 
null-points D. 
3. The null-curves (D)p and (Dig have the 6(m—1)* singular 
points in common; for, a critical point bears oo! pairs d, d’. 
The two curves pass further through the (42—7) null points 
of the straight line PQ. Each of the remaining intersections is a 
point D, for which d passes through P, d’ through Q. In other 
words, if d revolves round P,d’ will envelop a curve of class 
