of the lines s being the second directrix. It is quite determined by the 
lines s and the transversal out of #, via d,, and ¢. 
B. In the plane (s,,s,') lies a pencil (d®), having /’, and the 
intersections of s,, s, and d,, as base-points. Each of these J? forms 
with d,, a figure 9’. 
In the same way d,, is to be combined with a d* of a pencil 
lying in the plane (s,, s,') 
C. The straight line ¢ may be coupled to any conic J? passing 
through 4, F,, F,, which meets ¢. 
D. In the plane (/’,s,) lies a (d*), having as base-points /’,, F, 
and the intersections of s,,s,'. The corresponding straight line d 
passes through /’, and rests on s,'. Both component parts of (d, d®) 
are variable. 
In the same way each of the planes (/',s,'), (/;s,), (/,s,') contains 
a pencil (d*); the corresponding pencils of rays lie in the planes 
Hs,), A75), GAP 
Here too the figures d° form a locus of order ten. 
The intersection of the plane /’, FF, with the surface 4 now 
consists of a conic (resting on /), the straight lines /’,/’, and F,/, 
(belonging to figures Jd’, .intersecting / elsewhere) and the straight 
line /,/',, which is a triple one, because 4* contains three d* resting 
on /. The straight line / consequently determines a surface /’, 
which has d,, as triple ae line, passes through d,,, d,,,/ and 
possesses four nodal lines s,, 5,', 53,51; Fo, f, are quintuple points, 
FP, is a triple point. 
In an arbitrary plane ® this congruence determines a cubic in- 
volution with one singular point of order three, four singular points 
of order two, and three singular points of order one. *) 
4. ie shall finally consider the Le” |, which has /’,, /’, as fun- 
damental points, the straight lines s,,s,',5," and s,,s,',5," as singular 
bisecants; the first three meet in F. the other three in /’,. 
The straight line d,, = /',/’, is triple directrix of a ruled quartic 
surface A*, which has the six straight lines s as generators. Any 
plane passing through two generators intersecting on d,, intersects 
4* moreover along a conic d* resting on d,, and on the straight 
lines s, consequently forms a degenerate figure 9’. 
In the plane (s,s,') lies a pencil (d*) having as base-points /’, and 
the intersections of s,,s,',s,"; each of these curves forms a figure 9° 
with a definite ray d of the plane pencil which has #, as vertex 
1) It has been treated more fully in my paper quoted above (Proc. XVI, S 13). 
