$82 
found as follows. A plane through any point A, of the generatrix a, 
and the corresponding generatrix 5, cuts both reguli respectively in 
a conic a,’ and a line 8, On these sections the other pairs of corre- 
sponding lines a,) determine two projective ranges of points (A), (5). 
As these arrange the rays of a pencil in the plane (Ah) in a 
correspondence (1,2), the lines AB envelop a rational curve of class 
three with ?, as double tangent. Each of the three lines AB passing 
through A, rests on two pairs a,b and belongs therefore to the 
congruence. 
The curve of class three just found and the pencil with A, as 
vertex form together the complex curve of plane (Ab). Likewise 
the complex cone of A, breaks up into this pencil and a rational 
cubic cone. 
Any point and any tangential plane of the quadratic scrolls (a), (0) 
is singular. Moreover the points of the principal planes and the 
planes through the principal points are s:ngular. 
§ 4. If we add the relation (p'r') = 0 to the conditions enumerated 
in § 2, it follows from 2(q') + (pr) =O that the coordinates gq’; 
also determine a line, which is to cut p' and 7’ on account of 
(pq) = 9, (qr) =0 without belonging to the regulus. So it lies 
either in the plane t through p’ and 7 or on a quadratic cone with 
the point of intersection 7’ of p' and ras vertex. 
In the first case each line of t belongs to the complex and even 
twice as it cuts two generatrices of the regulus (a). In other words: 
t is a double principal plane. 
In the second case an analogous reasoning shows that 7’ is a 
double principal point. 
$ 5. In the two latter particular cases the complex has lost the property 
of corresponding dually with itself. On the contrary’ this property 
is still preserved by the complex generated by two projective reguli 
the first of which consists of the tangents of a conic «° (in plane a) 
and the second is formed by the edges of a quadratic cone (with 
vertex 5). : 
The range of points B, on the section @,? of 8? and « is in (1,1)- 
correspondence with the system (a), So the points B, are in (2,2)- 
correspondence with the points of intersection A, of the generatrices 
a and the come @,?. So the complex admits four principal points, 
each of which bears a principal plane. 
Furthermore « is a double principal plane, B a double principal 
„point. 
