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vertices) whose vertex P is the focus of /, because the two conjugated lines 
of /, which cross each other in general and exactly therefore generate 
a regulus, now both pass through ? ; but P does not lie on 2°, 
because / is a ray of the complex, but not of the congruence. A 
generatrix of the cone therefore intersects 2", as formerly a line of 
the regulus, in six points, from which ensues that / now again is 
a sixfold line of the surface. And to a plane À through / corre- 
sponds as formerly a twisted cubic through the cone vertices and 
which now passes moreover through //, because / is a tangent of 
the complex conic lying in A, but which now again intersects 2°, 
except in the cone vertices, in fourteen points; thus in À lie 14 
generatrices of the surface, so that this is indeed of order 6 + 14 == 20. 
The curve £°, the section of the cone with @°, has also 6 nodal 
points lying on 4°, so that 2° contains 6 nodal generatrices. 
The nodal curve of $2°° undergoes a very considerable modification 
as regards the points it has in common with /. Through such 
a point namely must go 2 generatrices of the surface lying with / 
in one plane; but now / is itself a ray of the complex and three 
rays of the complex can then only pass through one point when the 
complex cone of that point breaks up into two pencils; so the only 
points which the nodal curve can have in common with / are the 
points of intersection of / with the four tetrahedron faces. 
These points which in § 15 we have called S; coincide with the 
points which were called 7;* in the same §. Let us assume the 
plane /7,. As now again and for the same reason as before nine 
of the fourteen generatrices of °° lying in this plane pass through 7 
(§ 13) the five remaining ones must pass through another point 7,* 
lying in t, and whose complex conic breaks up into rt, and the plane 
T,*l; now however this point coincides with S,. For the complex 
cone of S, likewise breaks up into two pencils, of which one lies 
in t,, the second in a plane through 7,* and 7, ; now however, to this 
second pencil evidently belongs our ray / and so indeed the complex 
cone of S, degenerates in this way into tr, and a plane through 
l; so S, and 7,* are identical. To S,, regarded as a focus, a ray s 
through 7, is conjugated which lies at the same time’on_ the 
quadratic cone, thus in other words the ray P; 7, ; the latter intersects 
2° besides in 7, in 5 more points and the rays s conjugated to 
these are the 5 generatrices of 27° through S,—= 7,* lying in the 
plane /7,; the sixth generatrix through this point conjugated to 
T, lies. in t,, but not in the plane /7',. 
So we see that through S, pass five generatrices of 2° lying in 
the same plane; so the four points S; are }.5.4=10-fold points 
