1255 
With the cone £* ZH has the curve gp?’ in common; the remaining 
section can only consist of straight lines. To it belong the three 
parabolic bisecants Pf), and the four singular bisecants 5. From 
this it ensues that the three trisecants d which ep? sends through 
P are nodal lines of HI’. 
For a point S of the singular curve o° the surface /7° degenerates 
into the monoid +* and a cubic cone &*, formed by singular bi- 
secants 6. The straight lines 5 form therefore a congruence of order 
four, with singular curve o*, consequently of class nine. 
7. The surface A formed by the 9%, intersecting a straight line 
l, has the 9° intersecting / twice as nodal curve. 
As 7 intersects every monoid ** thrice, s and 6? are triple lines 
on A. The section of 4 with the plane (/'s) consists of the triple 
straight line s and three conics @,’; of these, one passes through 
the intersection of /, the other two are determined by the two 
curves 9,° resting on /. So A is a surface of order nine, with triple 
points in FF, F;. 
On A’ lie 15 straight lines, 9 conies, 9 curves @* and 15 rational 
curves of. For / intersects 4 bisecants b, 11 curves 9’; 3 conics 
and 6 curves 9’. 
A plane À passing through / intersects //" along a curve 4°; the 
latter has in common with / the points, in which / is intersected by the 
0°, which has / as bisecant. In each of the remaining six points À 
is touched by a 9° of the congruence. 
The locus of the points in which a plane p is touched by curves 
0’ is therefore a curve p°. It is the curve of coincidence of the 
quintuple involution, which determines |9°| on g. The intersections 
S*,S,,.S,,S, of the singular lines s,o* are apparently nodes of g°. 
With the surface 4’ belonging to an arbitrary straight line /, ¢" 
has in those intersections 4 X 3 > 2 points in common; in each of 
the remaining intersections p is touched by a @° resting on /. 
The curves 9° touching p form therefore a surface D°. 
A monoid 2? has in the points S*, 5, 4 >< 2 points in common 
with °; on g° lie therefore the points of contact of 10 curves 9° 
of the monoid. From this it ensues that s and 6° are decuple lines of d°°. 
With the curve wy‘, belonging to the plane yw, #*° has, in the 
four nodes of w’, 4 > 2 X 10 points in common; in each of the 
remaining intersections y is touched by a 9’, which at the same 
time touches the plane gy. There are consequently 100 curves o°, 
touching two given planes. 
The plane p bas with °°, besides the curve of contact p° to be 
