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On the nodal curve y°, along which the ruled cubic I* is inter- 
sected by the plane «, the triplets of rays c determine an /*. The 
conics joining two sets of this /* with the double point D and 
another point B of y° have in addition to these points two points 
B', B" in common, not lying on y*. The sets of the /* are therefore 
cut out on y° by the system of conics with basal points D, B, B', B". 
Only the pair of lines DB, B'B" furnishes a set consisting of three 
collinear points. It appears from this that the plane « contains one 
line of the ruled surface (é), for the line t= B' B" does not rest on 
the three rays c of a triplet only, but also on the ray a conjugated 
thereto. 
Through A passes similarly one ray of (¢). Since a is still inter- 
sected by two additional transversals ¢, t', the ruled surface (t) is of 
the fourth degree. 
The remaining curve a? which (é* bas in common with @, sends 
four tangents through A. Hence (¢)* contains four double rays of 
the involution (tf). 
If ¢ is caused to describe a pencil (7’,r) then (¢)* breaks up into 
(4) and a cubic regulus (¢')’. Now again «a contains one of the rays 
t'; the points of transit of the remaining lines ¢' constitute a conic 
«@, which passes through A and intersects t on the double rays 
which belong to the pencil. Hence the double rays of the involution 
(t,t) form a quadratic complex. 
9. Let a, be the particular ray of (A, a) which is intersected by 
the single directrix e of (c)’. Every line ¢’ which rests on a, is in 
(4, t’) conjugated to e. To the line t= e therefore correspond all the 
rays of a special linear complex. 
Similarly the double ray d of (c)° is conjugated to all the rays 
of the special linear complex having the ray ag which rests on d 
for its axis. 
In this involution (¢,?’) also the rays ¢4 through A are singular 
and each conjugated to the rays of a regulus having three lines c 
for its directrices and containing the lines d and e. 
Similarly the rays 4, lying in the plane a, are singular too and 
each correlated to the rays of a regulus which contains d and e. 
Now consider the system of the hyperboloids (//), which are each 
determined by three lines c. The specimens whieh pass through a 
given point P arrange the lines ¢ into the sets of a cubic involution 
of the second order. The involutions /?, which thus belong to the 
points P,P’, P", have one set in common; the hyperboloids /7 
therefore form a complex (triply infinite system). The hyperboloids 
