1254 
points B lie on a curve B'° with septuple secant s. In the same way 
it appears that 8'° meets 6? in 15 points. The surfaces *' and (b)' 
have in s and o* a section of order 4+ 3 X 2X3; moreover they 
have B'° in common. The remaining section of order 12 must consist 
of straight lines belonging to degenerate figures 9°, each composed 
of a o° and fwo straight lines 6 intersecting it. From this it ensues 
that [9°] contains sie figures consisting of a twisted cubic and two 
of its secants. 
This result may also be formulated in this way: through three 
points /;, pass 6 curves y* which intersect a given o* four times 
and a straight line s twice. Such a g° intersects the ruled surface 
(Dt in two points B lying outside s and o*; through these points 
pass the two straight lines >, completing 9’ into a 9’. 
N 
6. Any straight line d having three points in common with a 0° 
is a singular trisecant of the congruence. For through it passes one 
®* and the remaining surfaces of the net intersect it in the triplets 
of an involution. From this it ensues that the trisecants of the o° 
form a congruence of order three, as a 9° is intersected in each of 
its points by three trisecants. In § 3 it has been proved that any 
point S of 6° also sends out three straight lines d; on these singular 
trisecants, however, all the groups of the /, have the point S in 
common. 
Let 4 be a bisecant of a o° intersecting o*. Through it passes one 
p*; the net therefore determines on 6 an involution /?, so that 6 
is a singular bisecant. 
Through a point P? pass four straight lines b. For the curve @%,, 
which can be laid through P is projected out of # by a cone &*; 
the latter has in common with o* the eight points in which 9%, 
3 
rests on o®. The remaining four intersections lie on edges of £*, 
which have in common with o> two points not lying on o*, conse- 
quently are singular bisecants. 
These four straight lines 6 lie on the surface ZI, which is the 
loeus of the pairs of points, which the curves of [o°] have in common 
with their chords passing through the point ?. JM is apparently a 
surface of order six with quadruple point P, the tangent cone of 
which coincides with £*. 
I’ contains s and o*, therefore has with an aroditrary o° four 
points of s and eight points of o* in common; of the remaining 18 
points of intersection 12 lie on the 6 chords, which 9° sends through 
P, and 6 in the points /. Hence H* has three nodes Fr. 
