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plex is linear, for through any triplet of points JF} passes the sur- 
face corresponding to the point of intersection of the null-planes 
Pis Par Ps: 
The verter of the singular cone S, bears oo! singular null-planes 
6 not passing through a line; from this ensues that it coincides with 
the principal point T. 
In an analogous way the plane t of the singular conic o? is prin- 
cipal plane of the nult-system. 
Let us consider the plane through 7’ and one of the axes s*; 
it has for null-point the singular point S lying on s* but at the 
same time the principal point 7’; so it is singular and its null-points 
lie on the line s, = TS. So the regulus s, is a cone and consists 
of the edges of the cone projecting the singular conic o* out of 7. 
Likewise the axes s* form the system of tangents of a conic lying 
in the principal plane vt. 
$ 5. The conics (/)? form a system oo* admitting a representation 
on the lines of space. For through any two points /,, 2, one (/)? 
passes, which is completely determined by the line 7 common to 
the null-planes @,, Pz- 
The cones {/], each of which is the envelope of the null-planes 
of the points of a line / also form a system o*; any of these cones 
can be determined by means of two planes ¢,, y, the null-points of 
which indicate then the line /. 
If / lies in a singular null-plane o, the conic (/)? breaks up into 
the line s. bearing the null-points of 6 and a second line / which 
is bound to cut s,; so the principal point 7’ which also can figure 
as null-point of o cannot lie ontside sy. So we find once more 
that the regulus (s,) is a cone. 
If / passes through the vertex of >, and bears therefore two sin- 
gular planes, (/)? degenerates into two intersecting lines, the point_ 
of intersection coinciding evidently with the vertex of =,; for the 
null-point of any other plane through / must coincide with that vertex. 
$ 6. A special trilinear null-system is determined by the 
tangential planes of a pencil of quadratic surfaces ®* touching each 
other along a conic 6% where the point of contact forms the 
null-point. *) 
1) In the case of a general pencil with a twisted quartic as base we get a 
null-system (1, 3,2), treated at some length by Dr. J. Worrr (“Ueber ein Null- 
system quadratischer Flächen’’, Nieuw Archief voor Wiskunde 1911, vol. IX, page 85), 
