i 715 ) 
thus for the tangents out of D to the polar conic if 
4t/ 3 [eye J — y.x,x 3 — y2x1x3 -f y 3 (** _ *,^jj — (2^*3 _ y,* a _ y , Xx f. 
When one of these tangents passes through Y we have 
4 V3i e ] ~ %iW3 + y®) = (2 y\ — 2yty 2 )% 
or 
y-yl 4- vmy\ — = o. 
From this is again evident that the curve indicated by this equation 
is the locus of the points associated with respect to c 3 and collinear 
with D. 
This special nodal d* is characterized by the property according 
to which it. is touched by a cubic in the six points whose tangents 
concur in the node. For, when considering the pencil which 
is determined by d* with the conic of Bertini counted twice it is 
immediately evident that the remaining points of d 4 lying on this conic 
are points of contact of a double tangent, which must then also lie 
on the nodal tangents. 
8. We shall now see into what a line (D is transformed by the 
correspondence of the associated points. To that end we eliminate 
yk out of the three equations 
= 0, ay a? = 0 and a® a x = 0. 
Out of the first two we find 
y 1 • yz • yz (®2 13) a J : (« 3 §1) a* x : (<*1 fe) aK 
Substitution in the third then produces 
A line § is thus transformed into a curve § s of order five. This 
could be foreseen, for the two associated points lying on § pass 
in the transformation into each other, whilst the three points of 
intersection of % and c* correspond to themselves. 
When the point U describes the line §, its polar line u envelops 
the poloconica § s , whilst its polar conic u 1 describes a pencil. From 
this ensues that is generated by a pencil of conics and a pencil 
of rays of index 2 projectively related to it. Consequently § s has 
nodes in the four basepoints of that pencil and the points associated 
with U form the pairs pf the fundamental involution of pairs ap¬ 
pearing on §* x ). In connection with this I s is touched by the polo¬ 
conica §* in five points (1. c. p. 48). 
*) See my paper: “Ueber Gurven funfter Ocdnung mit vier Doppelpunkten”(Sitz. 
Akad. Wien, Bd. 104, S. 47). 
