714 
projecting cone having this point as vertex, and from this all follows 
easily that that edge of the cone is the line s conjugated to the point P 
of the nodal curve. The nodal curve now intersects £° in 24 points, 
of which 6 however coincide two by two with 7,, 7,, 7, ; the lines s 
conjugated to the 18 remaining ones are the lines of intersection 
of the cone of order nine with the doubly projecting cone at the 
vertex 7. 
The surface of tangents of k* is of order eight, it contains the 4 
just mentioned plane curves of order 4 as nodal curves and the four 
cone vertices as fourfold points; it intersects 2* in a curve of 
order 48 having the cone veriices as fourfold points, the 24 points 
of intersection with £° and the 4 times 18 points of intersection on 
the 4 nodal curves as nodal points. For an arbitrary point of this 
curve a chord a of &* and 2 chords 5 of &* are complanar; one of 
these two chords 5 however is a tangent of 4“. For one of the 24 
nodal points on 4° the same holds, as is easy to see; for each of 
the 4 Xx 18 remaining nodal points on the other hand a chord a ef 
k® is complanar to 2 tangents of k'. 
12. We now determine the second characteristic number, the 
class rv of the congruence formed by the rays s conjugated to the 
points of 2°, i.e. the number of rays of the congruence in an 
arbitrary plane. The locus of all foci of all the rays s lying in 
an arbitrary plane « is according to $ 10 a twisted cubic through 
the four cone vertices; this intersects 2° in 18 points, but to these 
belong the four cone vertices. To each of the 14 remaining ones 
One ray s is conjugated, lying in the assumed plane; to a cone 
vertex on the other hand all rays of the opposite tetrahedral face 
are conjugated, and therefore also the line of intersection of that face 
with a, so that if we like we can say that in each plane lie 18 
congruence rays, among which, however, then always appear the 
lines of intersection with the four tetrahedral planes. So we prefer to 
say that in an arbitrary plane te 14 congruence rays and that from 
the complete congruence the 4 fields of rays situated in the four 
tetrahedral planes separate themselves. 
In § 8 we found that the double tangential planes of the surface 
2,, discussed in $ 7 of class 18 envelop a developable A, of class 
9; they are nothing else than the focal planes of the points of 4°. 
The lines s they contain belong to the congruence we are discussing, 
and these rays count double in the congruence because £* is for 
2* a nodal curve; let us find the locus of these double rays. 
If a point P describes the curve #°, then each of its two polar planes 
