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481 
The pencils of planes, projecting the projective line-pencils (6), (c) 
from A, engender a quadrie cone (¢4)?, so A is a singular point of 
the congruence. Similarly the rays of the (3,3) which lie in the 
plane «, envelop a conic. Thus the congruence has four singular 
points of the second order (A, B, C, D) and four singular planes of 
the second order (a, 8, y, J). 
It also contains singular plane pencils of rays. If the lines a,b 
of a quadruplet lying on a hyperboloid meet, cand d intersect also. 
On a3 are situated evidently two points ab (coincidences of two 
projective ranges). Each of these is conjugated to one of the two 
points ed lying on yd. The transversals of four thus associated rays 
a,b,c,d form two plane pencils: one lying in the plane ab with 
centre at cd, the other in the plane cd with ab for its centre- 
Hence the congruence (3,3) has twelve singular plane pencils of rays”) 
8. If a pencil (¢) contains a ray of the (3,3), the ruled surface 
(’)’ breaks up into the singular regulus, to which the ray ¢ belongs, 
and a ruled surface (t) having A, B, C, D as double-points and 
a, 8,7, 9 as bitangent planes. 
If ( contains two rays of the (3,3), the locus of ¢’ consists of 
two singular reguli and a ruled surface (¢’)*, which has a conic 
and a line ?’ in common with a. The intersection of the aforesaid 
with r consists of two rays tt (passing through 7’) and a third 
line e, which constitutes the single directrix of (¢’)’; the double 
directrix passes through 7’. 
9. To the rays ¢ resting on a line 7 correspond the rays ¢t’ of 
a complex of the seventh order. In fact, / meets seven rays of the 
ruled surface (f)’, which is conjugated to a plane pencil of rays (t); 
this pencil therefore contains seven rays ?¢ of the complex into 
which the particular linear complex with directrix / is transformed. 
The complex {/’}’ has principal points at the vertices of the eight 
sheaves, principal planes in the eight planes of singular rays of the 
involution (¢,/’). For, in fact, / meets one ray of each singular 
pencil of the sheaf [A*| and two rays of a quadratic regulus (bed) 
passing through A; thus in the last case the corresponding ray t4 
is a double ray of the complex {t’}’. 
') The here considered (3,3) is a particular species of the congruence composed 
of the transversals of the corresponding rays of (hree projective plane pencils. 
This more general (3,3) too has 12 singular line pencils; contrarily, however, it 
has only 3 singular points and 3 singular planes of the second order. 
