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* and 6? lie on 
complex cone degenerates. Consequently the curves 9 
the singular surface of the complex. 
The edges of the ox! cones projecting the curves o,* from their 
corresponding point Zl, as vertices form a congruence of which we 
will determine order and class. 
The locus of the curves 6,* is the surface X'* into which the 
developable with g* as cuspidal curve is transformed by (P,3S). 
The cubic cones with an arbitrary point M as vertex and 0’ and 
o° as director curves, intersect in 9 edges, each of which connects 
a point S of 6,* with a point A’ of o°; if Fk’ coincides with the 
point RR, to which 6,* corresponds we have to deal with a ray of 
the congruence passing through J/. We will conjugate these 9 points 
R’ to R,. The line MR’ cuts the surface X'* mentioned above in 
12 points S lying in general on different curves 0,°; so to R’ 
correspond 12 points R. The correspondence (/,, 2’) has therefore 
21 coincidencies, i.e. the order of the congruence is 21. 
Any plane u contains 3 points A, and each of the corresponding 
curves 6,*° has 3 points S with gw in common; so the class is 9. 
So the lines S,R form a congruence (21,9) and an other congruence 
of the same type is formed by the lines SZ. The two congruences 
admit successively 0* and as singular curve. 
$ 5. Any point S* of the rational 6° (§ 2) is the vertex of a pencil 
of complex rays p the plane of which contains the corresponding 
tangent 7,. So the curves 9’ and c* lie also on the singular surface. 
The x‘ pencils with vertices S* form a congruence which we will 
study more closely. 
In any plane u lie 9 points S*; the tangents 7, corresponding to 
tiese points determine 9 rays p lying in u; so the congruence is 
of class nine. 
To any point S* we make to correspond the 9 points S’ of 6° 
which can be projected out of the arbitrary point M/ in a point of 
the corresponding tangent r,. The line MS’ cuts + tangents 7, so S’ is 
conjugated to 4 points S*. As any coincidency S’ = S* is due to a 
ray of the pencil with vertex S*, Af bears 13 lines RS*, i.e. the 
congruence is of order thirteen. 
So the complex contains feo congruences (13,9) each of which is 
built up of a? pencils. They admit successively 6’ and 9’ as singular curve. 
§ 6. To the complex (p) belongs the system of gencratrices of 
the developable determined by 9* and 6*. Any tangent 7, cuts four 
tangents s, and reversely; so the points of contact 7, and S, of the 
