881 
by. — Pl: Ne En 2q'k À fe rj. 
Then we find for the rays a of the congruence with director 
lines a, 6 ’ 
(pa) a? + 2(qa)44+ (re) =0, 
(pia) 2? 4 2 (qe) 2 + (re) = 0, 
which we abridge into 
PH+4+2QA4+R—0 , Pav+2Q004+R=0. 
By elimination of 2 we get the equation of the d¢quadratic complex 
under discussion. It is 
(PR! — PR) = 4 (PQ — P'Q) (QR — QR), 
or, what comes to the same, 
(re oe i) — 4 (Pm Oy (PR aor) 
From this ensues that the complex can be generated in two different 
ways by two projective pencils of quadratic complexes. This is shown 
by the equations 
PR —PR=?2u(PQ — P'Q), 
u (PR — PR) = 2 (QR — QR) 
and 
Pie Od =P ta (PRS GQ"), 
u (PR — 2QQ' + P'R) = 2(PR' — Q’). 
The equation (ab) =O expressing the condition that two corre- 
sponding generatrices a, 6 have a point in common, gives rise to a 
biquadratic equation in 4 So there are four principal points and 
four principal planes. 
§ 8. We now occupy ourselves with the congruence of the rays 
zw each of which rests on two pairs of homologous genêratrices (3). 
For such a ray w the two equations 
Pye tee oy PAS IOA AR 
must be satisfied for the same values of 4; so we have the condition 
Peg Ry | 
iat me | it | 
This equation leads to a congruence (3,3). For the quadratic 
complexes PQ’ =P'Q and PR'= PR have the congruence P=0, 
P’ =0 in common and the latter congruence does not belong to the 
complex QR’ = QR. 
This result is in accordance with the fact that the complex cones 
(and curves) must be rational and have to admit therefore three double 
edges (and three double tangents). 
Both the characteristic numbers of the congruence can also be 
