C= Saed), Ui Zar — a) (ey — Ys, 2)s 
3 3 
2 U = a} ET a5) (2, BY) eel) zie (x TA #,) @, Ul ale z)}. 
3 
The envelope of this system, at the same time the locus of the 
conics of the complex having / as chord, has for equation 
n= A T ne En 
U =U, Y= Ui): 
The eight nodes which this biquadratie surface of the comple. 
must possess are the points of intersection of the surfaces 
Te == iF —— 0} raes 
R=, ii Ore D= 0". 
For we have 
ÒU___ ÀU, OU, 
U + U, au, Ee 
DE a 3 3 
Ow ? 0a 
El 
= 
Ol 
so that — disappears for each of those eight points of intersection. 
Ow 
To these nodes evidently belong the points O, X,, Y,, 
four other ones change their places with the right line /. 
That / is double right line of the surface of the complex is 
immediately proved by the substitution «= 7, + 4a, y= y, + ue, 
Z=2,-+ v0; on account of 2,y — y,z =o (uz, — vy,) we see that 
U then obtains the factor o°. 
Ze; the 
Mathematics. — “On a group of complexes with rational cones 
of the complex.” By Prof. JAN pe Vries. 
§ L. In a communication included in the Proceedings of May 
1903 *), I have treated a group of complexes of rays possessing the 
property that the cone of the complex of an arbitrary point is 
rational. In the following a second group will be indicated with the 
same particularity. 
We consider a pencil (s) with vertex S in the plane o, and in a 
second plane tr a system of rays [f], with index m (thus the system 
of the tangents of a rational curve t,) and we suppose the rays ¢ 
to be projectively conjugate to the rays s. The transversals of homo- 
logous rays form a complex, which will be investigated here. 
Out of an arbitrary point P the pencil (S, 0) is projected on the 
plane t in a pencil (S', ©), projective to [¢],. Together these systems 
of rays generate a curve of order (n +1) having in S' an n-fold 
point; for on an arbitrary ray s through S' lies outside S' the point 
1) “On complexes of rays in relation to a rational skew curve.” VI, p. 12—17. 
