712 
Mathematics. — “On loci, congruences and focal systems deduced 
from a twisted cubic and a twisted biquadratic curve’. I. 
Communicated by Prof. Hk. pr Vries. 
(Communicated in the meeting of Oct. 26, 1912). 
11. We found in § 1°) a surface 2° as locus of the points P 
for which the chord a of #® and the two chords 5 of &* are com- 
planar; in the plane of those three chords then lies a ray s of the 
tetrahedral complex discussed in the preceding $*), so that the rays 
s corresponding to the points P of 2° form a congruence con- 
tained in the complex; we wish to know this congruence better. 
Through an arbitrary point P of space pass six rays of the con- 
gruence, thus w=6; for all rays s through that point form a 
quadratic cone, the complex cone (§ 10), and the foci correspond- 
ing to the edges of this cone lie on the ray s of P; this intersects 
2° in 6 points and the rays s conjugated to these pass through P. 
The number u is called the order of the congruence. J 
Exceptions we find only for the points of 4* and in the 4 cone 
vertices. If P lies on k° then the conjugated line s is the tangent 
in P, which now belongs itself to the complexcone of P, for it is 
generated as line of intersection of the two polar planes of P itself 
with respect to ®,, ®,, which planes coincide with the tangential 
planes to the two quadratic surfaces. The tangent s to £* is now 
however at the same time tangent to 2° and it contains therefore 
besides the point of contact only 4 points of 2°; thus besides the 
tangent only 4 rays of the congruence pass through P, from which 
ensues that the tangent itself counts double. 
The four cone vertices bear themselves quite differently. To 7, 
e.g. are conjugated as rays s all the lines of the plane 77,7’, = 17,, 
which plane intersects 2° in a curve 4° of order 6 containing 
TTT, as single points, the points of intersection with 4° on the 
other hand as nodal points; to each point of the curve a ray s 
through 7’ is conjugated, so that through 7, pass an intinite number 
of rays of the congruence forming a cone. This cone can be deter- 
mined more closely as follows. As of an arbitrary line s, in 1, the 
two conjugated lines pass through 7, the ray s, corresponding to 
the points of that ray s, form a quadratic cone; now s, intersects 
the curve £° in 6 points, thus the quadratic cone must intersect 
the cone to be found in 6 edges. 
Let us consider the point of intersection of s, with the edge 7,7, 
1) See Proceedings of Oct. 26th, 1912, p. 495. 
2) 1. ce. p. 509. 
