98 
is the image of a figure consisting of the straight line aj and a 
conic 97% in the plane (C’6z), which is intersected by az. 
On the curve y’*, along which ®° is intersected by a plane w, 
the pencil (9 determines an involution /*. 
If a tangent plane is taken for ys, w*® becomes rational; the in- 
volution /* has in that case four pairs in common with a central 
[*. To it belongs, however, the pair of points lying in the node of 
yw* and arising from the o*, which touches at w there. So there are 
three pairs of points that send their connectors through an arbitrary 
point. From this it ensues that the Aisecants of the curves 0? will 
form a cubic complex of rays, T*. C is evidently cardinal point of 
I’, for that point bears oo rays. 
The planes of the sav conics 90°, are cardinal planes. 
3. The rays of the complex passing through a point 7’ form a 
rational cubic cone, which has the straight line PC as nodal edge; 
for it intersects @* moreover in two points, so that it is chord of 
two 9°. 
The ends UU’ of the chords forming this cone lie on a twisted 
curve t°, which has a node in C; for any plane passing through 
TC contains apart from that edge only two more points U. 
If VC becomes tangent of *, the nodal edge passes into a 
cuspidal edge. The locus for the vertices of complex cones with a 
cuspidal edge is therefore the enveloping cone of ®*, which has C 
as vertex, consequently a cone of order four. 
For a point N on @&° the complex cone degenerates into the 
quadratic cone that projects the e* determined by .V, and a plane 
pencil of which the plane rv passes through C. 
Tf .V lies on one of the conics 0*;, the complex cone consists of 
three plane pencils, of which one lies in the plane of the conic, 
one in the plane (Na). 
If N is taken on one of the singular bisecants 4; the plane pencil 
(N, vj) consists of chords of 07,. 
In a plane v the complea curve degenerates into the plane pencil 
with vertex N and the twice to be counted plane pencil with vertex 
C; for a straight line passing through C is chord of two 9%. 
3 
4. The tangents out of NV at the cubic v*, which v has in common 
3 
with ®*, are at the same time tangents ¢ at curves 9°. This holds 
also for the straight line that touches rv? in C; but the latter, as 
ray of the congruence |f| is to be counted twice. 
From this we conclude that the class of [ft] is sae. 
