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the three straight lines d,,,s,,s,- These curves lie on the hyper- 
boloid H?, which contains the three straight lines mentioned and 
the points F,,/’,'). 
B. The straight line /',/’,=d,, may be coupled to any d?* of 
the pencil in the plane (/,s,), which has the points /,, /, and the 
intersections of s, and d,, as base-points. Similar systems of degenerate 
o* are determined by the straight lines d,,,d,,,d,, with pencils 
lving in the planes (Ps), #5), (/35,)- 
C. The transversal g, of s, and s, passing through /, may be 
coupled to any d? of a pencil in the plane £, #, #F,; the base 
consists of £,, #,, /, and the intersection of gs. 
Analogously with this is the system determined by the transversal 
g, of s,, s, through #,; the pencil lies in that case in the plane 
PF, 
D. In the plane F, F, Ff, a pencil (d°) is determined, the base 
of which consists of the intersection S, of s, and the points /,, /,, 
F,. To each d? belongs a ray d of the pencil which has #, as 
vertex and is situated in the plane (/’, s,). In this system both com- 
ponent parts of (d, d*) are variable. 
A system analogous with this is formed by the pencil of rays in 
the plane (Fs,), with vertex F,, and a (d*) in the plane #, HB 
Summarising we observe that the figures d* form a locus of 
degree ten. In the general congruence of the first principal group 
the figures Jd? form also a surface of order ten; it does, however, 
not consist, as in this case, of different figures. 
2. We can now easily determine the order of the surface d 
formed by the gy’, intersecting a given straight line / For that 
purpose we observe that the intersection of 4 with the plane /, £, /, 
must consist of figures d and d*. To this belongs in the first place 
the J? of the pencil lying in this plane, which meets /; further 
twice the straight line d,,, for / rests in its points of intersection 
with the hyperboloid H? on two 4’; finally the two straight lines 
dd, each belonging to a figure the d* of which rests on /. The 
intersection with FFF, is therefore a figure of order six, passing 
four times through #, and F,, thrice through #,. 
The curves 9’ intersecting / consequently form a surface A", having 
d,, as nodal line, passing through dd, d,,,d,, and possessing 
1) The conics passing through two points and resting on three arbitrary straight 
lines form a quartic surface. Here the planes #3 F, F, and #3 F, Fj contain each 
a pencil of conics which cannot be taken into consideration so that their planes 
fall away. 
