273 
We shall now determine the class of the curve enveloped by the 
straight line Qt: it is at the same time the order of the line- 
complex formed by the bisecants of the curves 9°. 
To this purpose we make use of the following general proposition *) : 
If a curve ¢” is intersected by a pencil (4#) in the groups of an 
involution /s, the lines connecting the pairs envelop a curve of class 
4(n—1)(2s —n). 
From this it appears that the bisecants of the curves 9° form a 
complex of order nine. 
§ 38. We arrive at the same result by paying attention to the 
pairs of lines of the pencil (4*). The straight line G,G, determines 
by its intersection Q with f, a »*, which meets the straight line 
G,G, in three points of a curve g°; hence G,G, is a trisecant 4. 
In the same way G,G, and G,G, appear to be trisecants. These 
three straight lines evidently replace nine bisecants. No bisecant can 
belong to the plane pencil (Gy) as it would have to lie then on 
a pair of lines of (4°). From this it ensues that the complex of the 
bisecants is of order none. 
Im a plane passing through the straight line c, lies, as appeared 
above, a plane pencil of trisecants. As all the g° pass through the 
points C,, C,, any ray passing through one of these points, is bisecant 
for three different curves 9°, which are indicated by the iutersec- 
tions of that ray with #*. In any plane passing through c the 
complex curve degenerates therefore into three plane pencils, which 
must each be counted thrice. 
The complea cone of an arbitrary point P has three triple edges. 
One of them is the ray, which the congruence (1,3) of the trisecants 
sends through P, the other twe connect P with the cardinal points 
CG, 
For a point of the straight line c the complex cone is replaced 
by the rational cubic cone, which projects the curve y*; this cone 
consists completely of trisecants and is therefore to be counted thrice. 
If P is taken on #*, the complex cone degenerates into the cone 
of order four %*, which projects the curve g° indicated by P, 
consequently has a nodal edge, and a cone of order five X*, which 
is the locus of the sets of four bisecants which the curves 9° send 
through P. 
1) Cf. my paper ‘Quadruple involutions on biquadratic curves”. (Proceedings 
and Communications of the Royal Ac. of Sc. section Physics, series Ill, volume 4, 
p. 312. French translation in Archives Neerlandaises, Vol. 23, page 98). 
18 
Proceedings Royal Acad. Amsterdam. Vol. XIX. 
