580 
The double edges of a complex cone are rays resting on two pairs 
a,b; they belong to a congruence contained in the complex, of which 
congruence both order and class are equal to the number of double 
edges of the cone. . 
Evidently any point common to two corresponding generatrices 
a,b is a principal point, the plane containing these lines a principal 
plane of the complex. If one of the. scrolls is plane, the bearing 
plane is a principal plane too; if one of them is a cone, the vertex 
is a principal point *). 
Any point P of a principal plane is singular, the pencil with 
vertex P lying in that plane forming a part of the complex cone of 
P. The same degeneration presents itself for any point of each of 
the given scrolls; so these surfaces are loci of singular points. Like- 
wise any plane through a generatrix « or 4 and any plane through 
a principal point is sigular. 
By means of one scroll only can also be obtaine complexes con- 
sisting of linear congruences. So we can arrange the generatrices of 
a scroll in groups of an involution / and consider any pair of any 
group as director lines of a linear congruence *). 
In the following lines we treat the Abiquadratic complex which 
can be derived in the manner described above from two projective 
reguli. After that we will investigate the particular cases of plane 
scrolls or cones. 
§ 2. We use the general line coordinates a, introduced by Krein, 
which are linear functions of the coordinates p of Prücker and 
satisfy the identity (2?) = = a,’ = 0, while = Pie Ye 0. OF (cy) ae 
6 
indicates that « and y intersect each other. 
Then a regulus is characterized by the six relations 
ay, = pea? + 2qe 4+ Th 
satisfying the conditions: 
(p?) = 0, (r*) = 0, (pq) = 9, (qr) = 9, 2 9") + (vr) = 9. 
Likewise we represent the second regulus by 
1) In our paper “Sur quelques complexes rectilignes du troisième degré’ 
(Archives Teyler, 2nd series, vol. IX, p. 553—573) we have considered among 
others the case that one of the scrolls is a pencil whilst the other is formed by 
the tangents of a conic. 
2) This has been applied to a developable in our paper “On complexes of rays 
in relation to a rational skew curve’ (Proceedings of Amsterdam, vol. VI, p. 12) 
and on a rational scroll in “A group of complexes of rays whose singular sur- 
faces consist of a scroll and a number of planes”. (Proceedings of Amsterdam, 
vol. VIII, p. 662). 
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