Thy 
This point is a certain point of the line of intersection of the plane 
IT, with the face vr, lying opposite 7’; for, for an arbitrary point 
P of rt, the complex cone breaks up into the pencil with vertex P 
lying in v, and a pencil with vertex P lying in a certain plane 
through P and 7’, and inversely for a plane through 7), so e.g. 
our plane /7,, the complex conic breaks up into point 7, and a 
second point lying on the line of intersection of that plane with z, 
(§ 10). So the four singular points on / are therefore nothing else 
but the points of intersection with the four tetrahedral planes. 
Two congruences according to the theorem of HaLPHEN possess in 
general only a finite number of common rays; however, the con- 
gruence discussed above and the one deduced out of the points of 
2° possess an infinite number, therefore a scroll; for all complex 
rays s cutting 7 belong to the former, and every time 6 of these 
through a point of 7 belong according to $ 4 to the second; the 
two congruences have thus a scroll in common for which the line / 
is a sixfold line. As furthermore according to §12 there lie in each 
plane 14 rays of the second which as rays s cutting / also belong 
to the former (and therefore, as we now discover, envelop a conic) 
the scroll to be found is a @° of order twenty and with a nodal 
curve which by each plane through the sixfold line / is eut in 
1.14.13 = 91 points not lying on /. 
If a point P describes a line /, then the corresponding line s 
describes a regulus through the 4 cone vertices ($ 4); and if we 
wish to construct for that same point P the complex cone, then 
according to § 10 we must determine the lines s which correspond 
to the points of the line s conjugated to P?; from this ensues that 
the regulus formed by the lines corresponding to the points P of t 
ds the locus of the points P whose conjugated rays form the congru- 
ence of the rays s which intersect |. And so furthermore from this 
ensues that the curve k'? of order twelve alona which that regulus 
and 2° intersect each other, is the locus of the points P whose 
conjugated lines s form the just found surface 2*°. 
Each generatrix s of the regulus contains 6 points of 2° or there- 
fore of k'°; the corresponding lines s are the six generatrices of £2°° 
issuing from the focus conjugated to the generatrix P of the regulus 
on the sixfold line 7. The curve 4" admits 6 nodal points, viz. the 
points of intersection of the regulus with /*; the line s of the 
regulus through such a point intersects @° in two coinciding points, 
from which ensues that through the point P on / conjugated to that 
line s really only 5 rays s pass instead of 6; one of these, however, 
viz. the one corresponding to the nodal point of 4, is a generatrix 
