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PQR. For (R)' meets the curve (/)* belonging to 7—= PQ in eight 
points on o° and therefore in one point outside o°. 
Evidently o° is base curve of the linear complex of surfaces GE 
§ 8. A special null-system (1,1,2) can be obtained in 
the following manner. We start from two pairs of non intersecting 
lines a,a’ and 6,6’. We assign to any point F' the plane » of the 
two transversals ¢ and w through # over a, a’ and 3, b’. 
The hyperboloids (/aa’) and (/bb’) admit a curve (/)* of which 
/ is a chord as completing intersection. So we have indeed y= 2. 
Also a, a’, 6, 6’ are chords of (l)°. 
Here the singular curve 6° is represented by the lines a, a’, b, b’ 
and their gwadrisecants q,q’. So the figure of singularity has eight 
points in common with (/)'. | 
For any point S of a the transversal u is determined while we 
can assume for ¢ any ray of the pencil (Sa’). So the null-planes of 
S form a pencil with axis u. So the scroll (s*) breaks up here into 
the four reguii with the director lines (a, 6, 6'), (a', b, b'), (b, a, a’), (b', a, a’). 
For any point of q the transversals ¢ and w coincide and the same 
happens for any plane through q'. So the lines q,q' are not only 
loci of singular points but also envelopes of singular planes. As this 
is also the case with the lines a, a’, 5, b' the two dually related figures 
of singularity are united. 
§ 9. For a line / intersecting a in A the locus (/)* breaks up 
into a conic (/)? and a line w containing the null-points of the sin- 
gular plane (la); the conic lies in the plane (Aa') and passes through 
A, this point being the null-point of the plane connecting / with the 
transversal wu, through A. 
If / meets q, the curve (/)° degenerates in g and an (J)*. The lines 
/ determining conics (/)* form therefore sz special linear complexes ; 
so there are oo? conics (/)’. 
If / meets both lines q and q' the hyperboloids (laa!) and (/bb') 
intersect in /,q,q' and. a fourth line / meeting q,q as l does. So 
the relation between / and /' is involutory ; each of them contains 
the null-points of the planes passing through the other, the planes 
containing either g or q’ discarded. 
If / meets a and 5, the curve (/)° breaks up into a line u in the 
plane (al), a line ¢ in the plane (b/) and a line / cutting ¢ and u 
containing the null-points of the other planes through J. 
If we assume for / a transversal /, the curve (J)* is represented 
by the lines w and w' of the planes (al), (a'l) and by ¢ itself. This 
