( 714 ) 
conics with index 2 and a pencil projective to it is characteristic 
for the nodal biquadratic curve 1 }. 
6. Each nodal biquadratic curve d* of which the nodal tangents 
pass through the points of contact B and B' of a double tangent 
d is related in the way mentioned above to a c 3 . 
The polar curve d 3 of D has in B the tangents t and f in common 
with d* and it intersects it in the points of contact R of the six 
tangents concurring in B. Of the 16 points which d A has in common 
with the system of d 3 and d six lie in B, four in B and B' , six 
in the points R. The tangents t and f contain eight of those points; 
so the remaining eight lie in a conic (curve of Bertini). 
This conic d' unites the six points R to the points D and D\ 
Let us now regard the pencil determined by d* and the conic dr 
counted twice; one consisting of the double tangent d and a 
.cubic c 3 belongs to it. From this ensues that d A is touched by c* 
in the points of contact R of the tangents drawn out of B to d\ 
As dr passes through the points R, it is the polar conic of B with 
respect to c 3 ; because B and D" are the fundamental points, so 
that BB and BB’ are touched by d 2 in B and B\ d is the polar 
lme of D with respect to d' and of c 5 . So d 4 is the locus of the 
points associated with respect to c* and collinear with D. 
7. If d* is represented by 
x \ x \ -f «i * 2*1 — c 3 ^ = 0, 
where 
^ ~ ( c i®i 4- C2«?a)( 3 ) 
then t\ f, and d are indicated by * x = 0, x % = 0, and = 0, and 
d 3 by 2x l x 1 x l — = 0. From 
(®? x l + XiXlX \ ~~ <£**) — « 3 (2*u**s — <£) — 0 
then follows for d 1 the equation 
s= 0, 
and from 
<*“'* - *? - <*•») = 0 
we find for c* * 
~ -b aP = 0. 
For the polar conic of Y with respect to c> follows from this 
V i - S' 1 * 2 * 3 - 1*3 + y> w — = 0 , 
for the polar line of D with respect to y 
-2/,*, = 0, 
• in Wien, Bd. 53, S. U9. 
v ) b °bek, Denkschriften , 
