592 
to these. Through a linear transformation of the plane a of the 
conic 4? we can transform the points A, and A, into the circle 
points at infinity. If S be an arbitrary point of the conic k?, the 
straight lines SP and SP’ will be harmonically separated by SA, 
and SA,,. hence they will be perpendicular to each other after the 
transformation, so that PP’ is a diameter of the circle 4?, the same 
as the line QQ’. The chords PQ and P’Q’ are therefore parallel 
and consequently intersect on the line A,A,. 
To an arbitrary ray ¢ through the point B corresponds, therefore 
a ray through the same point, so that to the congruence {t’| there 
belongs in the first place the sheaf itself. 
To the line A,A, corresponds a bilinear congruence of rays, also 
belonging to the congruence [¢’|, besides two fields of rays. 
Through the point B passes a cubic cone of singular rays of the 
congruence (1,3) considered in § 11. To each of these rays corres- 
ponds a plane pencil which projects a line d from a point Q’ of 
the corresponding conic &*. The point Q’ is associated to the second 
point of intersection Q of the ray with the conic £?. 
The cubic cone mentioned has the line A,A, as nodal generatrix. 
The two generatrices coinciding with A,A, belong to the two dege- 
nerate conics k? consisting of the line A,A, and of one of the two 
lines 6,,6,; hence the two leaves of the cone A *, which pass through 
the line A,A,, touch at the planes of these degenerate conics conse- 
quently they also touch the two leaves both passing through A,4,, 
of the surface of order four, found in § 2, described by the conics 
k?; the line A,A, belongs therefore siv times to the intersection of 
the cone K* with this surface. The rest of the intersection consists 
of the curve o* projected by the cone KX? and of the locus t° of 
the points Q. The cone A®* has three points in common with each 
of the lines a,, a, and a, lying on the quartic surface mentioned ; 
the curve 6? having these lines as bisecants, two of these points lie 
every time on the curve o*, while the third must lie on the curve t’. 
It is further easily found that the curves o* and zr’ lying on one 
and the same cubic cone, have tree points in common. In general 
through the involution (P.P’), to a cubic curve a curve of order nine 
is associated. However the curve t° having one point in common 
with each of the lines a,, a, and a,, three straight lines 2 belong 
to this associated curve and as it has three points in common with 
the curve o and for this reason contains three singular points D’, 
three lines d belong to it. The complete locus of the points Q' is 
therefore a Curve r‚*. 
The rays in question, associated to the generatrices of the cone 
