718 
of the surface of the double rays of the congruence ($ 12), and 
thus evidently a double generatrix of 2°. So: the 6 points of inter- 
section of 1 with the surface of the double rays of the congruence 
deduced from 2° are double generatrices of 8**. 
The curve /?? passes through the 4 cone vertices and the lines s 
corresponding to them fill the tetrahedral faces lying opposite; so we 
can ask how @°° bears itself with respect to those faces. We now 
have separated in § 12 of the complete congruence deduced from 2° 
the four fields of rays in the tetrahedral faces; if we thus follow 
k” through the vertex 7’, then to all points on either side of 7, every 
time a completely determined ray intersecting / is conjugated; by 
this also in zr, one ray is determined, so that 2°° has simply one 
ot its generatrices in t, and therefore this plane as an ordinary tangential 
plane. The cone vertices on the contrary are themselves singular 
points ot 27°. Our curve &'? namely cuts rt, in 12 points lying 
on a conic and at the same time on the section 4° of 2° with r,, 
to which belong the three cone vertices 7',, 7, 7; the rays s corre- 
sponding to these lie, it is true, according to the above, respectively 
in tT, T;, T,, but they do not pass through 7’ (if let us say s conjugated 
to 7, had to pass e.g. through 7, it would have to pass for the same 
reason through 7, and 7), however the rays corresponding to the 
remaining 9 points of intersection do; so in the plane 7,/ nine 
generatrices of 27° pass through 7); they are the lines of intersection 
of this plane with the cone of order nine, on which le according to 
§44 the rays s which are conjugated to the points of intersection of 2" 
with t,. The same holds of course for the planes through the reraain- 
ing vertices and /. 
In such a plane the conic which must be touched by the 14 
generatrices of 27° degenerates, as we have seen at the beginning 
of this §, into a pair of points; so in each of these four planes not only 
nine generatrices pass through a cone verter, but also the five remain- 
ing ones pass through another fiwed point, lying in the opposite face. 
The vertices are thus for the nodal curve of 27°} .9.8=36-fold points, 
the other points }.5.4==10-fold points. If we add these 36 + 10 
points to the 45 points generated by the intersection of the two groups 
of 5 and respectively 9 generatrices lying in a plane through a cone 
vertex and /, we find back the 91 points of the beginning of this $. 
If we add to the figure, as we are now studying it, another arbitrary 
line m, then to this also belongs a regulus through the 4 cone vertices 
cutting the regulus conjugated to / in a curve of order four through 
the vertices; this biquadratic curve has with 2° twenty-four points 
in common among which again the cone vertices; if we set these 
