1164 
the n rays of the congruence lying in t. As in § 10 the line s these 
n rays partake of the property that the null-plane of any of their 
points is indefinite. 
The singular null-planes touch in the points of o? the scroll [¢)?"+? 
with 6%, a”, a as director lines and ” double generatrices in 1. The 
polar plane of P cuts o* in 2(2n-++1) points each of which bears a 
singular null-plane; so the singular null-planes envelope a torse of 
class (4n + 2). 
Evidently r is once more principal plane. 
The bisecants of a twisted cubic «° determine in an analogous 
way a null-system (1,1, 4). Here each point S of the singular curve 
a? is vertex of a quadratic cone enveloped by the null-planes of S. 
Now two surfaces (P)’ have in common the singular curve a’, 
to be counted four times, the singular conic 6%, a curve (l)° and 
finally the tnree chords of «° lying in t. 
§ 12. By the considerations of § 11 we have shown that bilinear 
null-systems with y > 2 do exist. 
Now we will prove that the locus of the singular points of a 
null-system (1,1, 7), with the condition y > 2, cannot be a single 
curve. 
Evidently the curve (J’+' containing the null-points of the planes 
through / is rational, / being a y-fold secant. The null-points of the 
planes through P lie on a surface (P)/+' touched in P by the 
null-plane of P. 
The surfaces (P)+! and (Q)/+' have a curve (/)/+! in common. 
Now let us suppose that the completing intersection is a curve 6 
of order y (y + 1). 
In order to determine the number of points common to (/) and 6 
we first determine the number H of transversals passing through 
any given point O and resting on (/) and o. ; 
For this number the known relation 
m(u—1)@~—1)=2h4+H 
holds, where u,v are the orders cf both the surfaces, whilst m is 
the order of the first curve and / the number of its apparent double 
points. 
Here we have u=v=m=yti, 2h=y(y— 1), as (J) is ratio- 
nal. So we get H=y (y? + 1). 
The transversals under consideration are common edges of the 
cones projecting (/) and 6 out of U; the remaining common edges 
pass through the points of intersection of both the curves, 
