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The tangent plane of the hyperboloid (6,c,c’) at A intersects a 
along a line « which touches (6, c,c’). The transversals of the four 
rays a,b,c,c’ therefore coincide. Hence every ray ta is also to be 
regarded as a double ray; thus the cubic complex of double rays 
has principal points at A and B and, accordingly, « and B as 
principal planes. 
It follows from the above, that the Snakes of lines A and B and 
the planes a and 8 consist of singular rays of the inwolution (t, t’). 
Together with A a ray 6 determines a pencil (¢4) and thereby at 
the same time a quadratic involution J? among the rays of the | 
regulus (c)’. Now, let there be given in the plane 4 a pencil (/) with 
vertex L; then each point of 82 determines, by means of /?, an 
involution 7? on the conie (17, 2). Through Z therefore passes a ray 
Ll joining the points of transit of two rays c, ce’, which in combination 
with 6 determine a transversal ¢4. If this ray / is conjugated to the 
ray U, which meets 6, a projectivity is established in (J). Each of the 
two united-rays is then a ray € which is conjugated to a ray ty. 
It follows from this that the reguli (¢’)? which are conjugated to 
the singular rays t4, constitute a quadratic line-complex. 
Three other quadratic complexes {t’}? correspond to the sheaf of 
lines [tp] and to the plane systems of rays [t,] and [4]. 
The pencil (7,1) contains two rays of each of these complexes; 
accordingly A, B,@, and B each carry two rays tf of the ruled 
surface (¢’)° into which (¢) is transformed by the involution (¢, 4’). 
Thus it appears again that (¢’)? has A and B as double points, 
a and B as double tangent planes. 
The ray AB meets two definite rays c‚c’‚ but all rays a and 6. 
To t=AB therefore are conjugated all the rays of the bilinear 
congruence which has ¢ and c’ as directrices’). Similarly {ag is 
conjugated to a7 rays ¢’. Thus the involution (¢, ¢’) has two principal 
rays, AB and af. 
4. The lines of the regulus (y)? too are principal rays, for a line 
y meets two definite rays a and 6, but all rays c; each transversal, 
t’, of a and 6 rests on two rays c and is therefore conjugated to 
= 
The involution (¢, t’) has still other singular rays. If the point of 
intersection, S, of two rays a and c lies in the plane o passing 
through two rays 6 and c’, then the pencil (S, 5) consists of rays s 
1) The congruence [t’], conjugated to AB belongs to the intersection of the 
line-complexes which correspond to the sheaves [¢4] and [tg]. 
