635 
must needs have a double point at A, as an arbitrary ray a is met 
by two transversals ¢,?¢’ only. Since the line AB outside A and B 
meets two lines ¢*) and therefore at A has two points in common 
with (@)°, it is necessary that A, and B too, is a double point of the 
ruled surface. 
The curve @* has six tangents passing through A; hence (£)° con- 
tains six united-rays (double rays) of the involution (¢, ¢’). 
The transversals of the pairs a,6 form a quadratic line-complex ; 
for, in an arbitrary plane (a) and (b) determine two projective point- 
ranges and the joins of corresponding points envelop a conic. This 
complex has four rays in common with the second regulus (set of 
generators) (y)? of FT“. Each of these tour rays meets two corre- 
sponding rays a,b and at the same time the rays c,c’ conjugated 
thereto. Hence the ruled surface (£° has four lines in common with 
the hyperboloid I’. 
2 If ¢ is caused to describe the pencil (7’,r) the ruled surface 
(D° breaks up into this pencil (¢) and a-ruled surface (t’)’. Thus the 
transformation (t, t’) converts a pencil into a ruled surface of 
degree five. 
Of the two united-points of the projective point-ranges on «8 one 
now lies at apr; through the other passes a ray ¢’. Thus in « (and 
in 2) there lie again two rays ¢’. The remainder of the intersection 
of 4’) and « is a nodal «° with double point at A. Each point of 
intersection of «° and r is the transit of a ray ¢’ which coincides 
with its conjugated ray ¢. Hence the double rays of the involution 
(t,t) form a cubic complex. 
A confirmation of this enunciation can be obtained as follows. 
With 7? (t’)’ has four rays y in common (§ 1) and in addition 
thereto a twisted curve y°. At a point of intersection, C, of y° and 
tT a ray ¢ is intersected by the corresponding ray ¢t’; hence C lies 
on a double ray t= and the second line of (c)* resting on this 
double ray meets rt in a point C’, which must lie also on y*. Thus 
the six points of transit of y’ lie in pairs on three double rays 
belonging to (Tr). 
3. A ray tú through A is intersected by a ray 6 and by two 
rays c,c’ of (c)*. Bach ray 4 which meets 6,c and ec’ intersects on 
a a certain ray a and is therefore conjugated to ¢4; hence the ray 
tg is singular. 
') Lying in the united-planes (coincidences) of the projective pencils of planes 
which project (a) and (hb) from AB. 
42% 
