1163 
For any point of s the cone projecting 5° degenerates into the 
plane t counted twice; so its null-plane is indefinite and this explains 
why s must lie on each surface (/)’. 
Indeed the plane rt is principal plane; for the null-plane of any 
point of t lying neither on 6? nor on s coincides with rt as polar 
plane of a line ¢ not situated in rt. 
In connection with this result the cubic torse of the null-planes 
of the points lying on / always contains the plane rt, i.e. t is common 
tangential plane of all the surfaces of class three enveloped by the 
null-planes of the points of a plane. 
The trace d of a singular plane must be incident with the pole 
D, i.e. it must touch 5°. In this case ¢ is transversal of a, a’, o? and 
each of its points may figure as null-point. The locus of these trans- 
versals is a quartic scroll |t\* with a and a’ as double director lines 
and the line s mentioned above as double generatrix. 
The polar surface of any point P with respect to [t]* intersects 
6 in six points; the planes touching [t|* in these points are singular 
null-planes. So these planes envelope a forse of class sia. 
§ 11. In the null-system considered in the preceding article the 
transversals ¢ form a bilinear congruence. If we replace it by a 
congruence (1,7) we get a null-system (1,1,2-+ 1)’). If the plane 
p rotates once more around the line /, in which case its trace d 
describes a pencil in t and the pole D a line 7, then the ray ¢ 
resting cn 7 describes a scroll of order n+ 1. So the null-point of 
y lies (2 +1) times on / (y=n +1) and describes a twisted curve 
(in, 
Let the congruence (1,2) be determined by the director curve a” 
and the director line a, which is to have (n—1) points in common 
with a”. 
Each point of a” is singular and bears a pencil of null-planes 
(see § 10). From a point of a the curve a” is projected by a cone 
of order n with an (n—1)-fold edge a. To the trace of this cone, 
considered as locus of D corresponds a curve of class 7, the envelope 
of the trace d of the null-plane vy. So each point A of a bears oo! 
null-planes enveloping a cone of class ”. So a is an n-fold line 
on the surface (P)"+?. 
Here also any point of the singular conic c? bears a pencil of 
null-planes, the axis of which touches o?. 
The intersection of two surfaces (P)"+? breaks up into a curve 
(/)"+?, the curves a” and o?, the line a (to be counted n?-times) and 
1) For n=O we get the null-system of § 6, for n=—1 that of § 10. 
