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of the tetrahedron. The two polar planes of this point now pass not 
only through 7’, but also through 7’,, because the point itself lies 
now not only in x, but also in vr, = 7,7,7,; so the quadratic cone 
contains the edge 7,7, and of course for the same reason 7’, 7’, and 
TT. These same edges lie also on the cone to be found and that 
as fourfold ones, which is easy to see when we consider e.g. the 
line 7,7,. This line intersects 4° in 7,,7’, and in four points more; 
to 7, all lines of rt, are conjugated and thus also particularly all 
lines of rt, through 7’, so that this plane (and for the same reason 
the two other tetrahedral planes through 7’) separate themselves 
from the cone; however, for each of the 4 remaining points of inter- 
section the conjugated ray s is determined and identical with 7), 7), 
so that this line is indeed for the cone under discussion a fourfold 
edge. So the quadratic cone and the cone under discussion have in 
common : 
1. the three fourfold edges of the latter, 2. the 6 rays s, conju- 
gated to the points of intersection of s, with 4°, thus altogether 
8 x<4+6—18 edges; so the cone under discussion is of order nine. 
If finally we see that this cone possesses three double edges too, 
formed by the rays s conjugated to the three nodal points of £° 
lying in £°, we can comprise our results as follows: 
For the congruence of the rays s- corresponding to the points of 2° 
the four cone vertices are singular points, as through these points pass 
instead of 6, as in the general case, w' rays of the congruence ; these 
form at each of thosc 4 points in the first place three pencils situated 
in the three tetrahedral planes through that point, and in the second 
place a cone of order nine with three double edges and three four- 
fold edges, the latter coinciding with the three tetrahedral edges through 
that point. 
The cone of order nine must intersect the tetrahedral plane 
t, = T,T,T, in nine edges, four of which lie united in 7,7,, four 
others in 7',7',, so that only one is left; the latter is to be regarded 
as the line s more closely conjugated to point 7’, and it will 
change its position if /° changes its form, and passes through 7’, in 
an other direction. 
The complete nodal curve of the surface of tangents of £* consists 
of four plane curves of order four lying in the four tetrahedral 
planes and every time with 3 vertices of that tetrahedron as nodes; 
let us now regard in particular the nodal curve lying in vr, 
Through a point P of this pass two tangents of 4* representing 
the two chords 4 through that point; the line connecting the two 
contact points passes through 7, and is an edge of the doubly 
