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line evidently contains the null-points of the remaining planes through 
t; therefore it is singular. 
We derive from this that the surface (P)* contains the transversals 
t and w passing through P; so the null-plane of P is a threefold 
tangential plane. The third line of (P)’ lying in that plane admits 
the property that the null-planes of its points envelop a cubie cone 
with P as vertex. 
If a,a', 6, 6' form a skew quadrilateral each null-plane touches one 
of the quadratic surfaces of the pencil with those four lines as base. 
Then the surfaces (P)* have four nodes in common, the vertices of 
the tetrahedron with a, a’, b,b',gq,q as edges. 
§ 10. We still examine an other null-system (1,1,2) the singular 
curve of which degenerates. 
Let us assume the conic o? in the plane z and a pair of non 
intersecting lines. Through / we draw the transversal ¢ over a,a’; 
then the polar plane of ¢ with respect to the cone /’(o*) may figure 
as null-plane of #. . 
Reversely, if the plane r is cut by p according to the line d and 
D is the pole of d with respect to o’, the transversal through D 
determines in p the null-point 7’. 
If p rotates around /, the line d deseribes a pencil around the 
trace R of 7 as vertex and D describes a line of r. But then ¢ 
deseribes a regulus with a,a’,7 as director lines, in projective corre- 
spondence with the pencil of planes (). Consequently the null-point 
F then describes a twisted cubic (/° with / as chord. The two points 
common to (/) and (/)* lie on the regulus. 
Each point A of the line a is singular. The transversal t describes 
a pencil in the plane (Aq’), its trace D with the plane r describes 
a line e bearing the trace A’, of a’. So the polar line d rotates 
round a point Z (pole of e); the null-plane of A describes therefore | 
a pencil with axis AL. 
If A describes the line a, the line e keeps passing through A’, and 
therefore E describes the polar line of A’,. So the axes of the pencils 
of null-planes corresponding to the singular points A form a regulus. 
A second regulus contains the axes of the pencils corresponding to 
the singular points A’ of a’. 
The conic o too is singular. Any point S of it admits as null- 
planes all the planes touching c° in S. 
All the surfaces (P)' have in common the singular curve o’, the 
singular lines a,a’ and also the line s through the traces A, and 
A,’ of a and a’ with t, containing two points S,,S, of of. 
