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dratie complex cone lying at infinitesimal distance lie of course also 
on the cone of order 9 (see above); so we can say more briefly 
that for each ray of this cone 7; is one of the foci and the tangen- 
tial plane to the cone is one of the focal planes. 
Each rav of the congruence through 7’, so each generatrix of 
the cone of order nine with this point as vertex, must have in P; 
two coinciding points in common with the focal surface; so 7; is 
for the focal surface a manifold point, however without the cone 
of order 9 being the cone of contact; for the tangential planes of 
this cone touch the focal surface in the foci of its generatrices not 
coinciding with 7; the cone of contact in 7; is enveloped by the 
focal planes of this last category of foci. 
21. Over against the question which complex rays through 7; belong 
to the congruence, is the other one which complex rays out of 7; belong 
to the congruence. In the preceding we have repeatedly come across 
these rays. Indeed, any surface 2°’ formed by the congruence rays 
which cut a line / or a complex ray s, and any surface 2"* formed 
by the congruence rays which cut a congruenve ray s contained 
such a ray as we proved above; we shall now show that all these 
rays form a pencil. To that end we imagine the tangential plane 
o in 7; to 2° and we cut it according to the line 7 by 7. We 
now saw in the preceding that the rays s conjugated to the points 
of tr, form a quadratic cone with 7; as vertex and containing the 
three tetrahedron edges through 77; ; if the base curve of this cone 
lving in 7; is 4*, then reversely the points of &* are the foci of the 
rays s lying in 9 and passing through 75, for the rays s conjugated 
to the points of a line pass through the focus of that line and 
the ray s conjugated to a point of tr; passes moreover through diss 
If a point P deseribes one of the rays of the pencil | 77] lying 
in o, say s,, then the rays s conjugated to the points P form the 
complex cone of the focus P, of s,, which point lies on &” ; this 
complex cone breaks up however into a pair of planes, viz 7; anda 
plane through P, and 7;, and the line of intersection ¢; of these 
two planes is the ray of the congruence conjugated to T;, insas tax 
as this point is regarded as a point of the ray s,; so the question 
is how the rays ¢; bear themselves when s, describes the pencil 
[7;| or, what comes to the same, how the planes 77 ¢; bear them- 
selves in those circumstances. We shall try to find how many of 
those planes through an arbitrary ray s, pass through 7; In each 
arbitrary plane through s, the complex conic breaks up into two 
pencils; one has the vertex 75, the other a point 7;* lying in 7. 
