( ) 
2. We shall see whether there are points U> for which the 
corresponding points V form again an associated pair, so that there 
is a triplet of points which are two by two associated. If we take 
the three points as vertices of a triangle of reference, their polar conics 
will be represented by: 
a A -f b x x 2 x t = 0, a 2 x 2 + b 2 x x *, = 0, a, x\ -f b t x, x 2 = 0 
for, each of those points has the connecting line of the other two 
as tangential chord with respect to its polar conic. 
If u = 0 is the equation of c*, the three polar conics are also 
represented by 
du ^ du du 
— = °, —~o, — = 0. 
dx x dx 2 dx 2 
From this ensues in the first place that the coefficients b lf b 2 , b t 
must be equal. Farther on it is directly evident that the equation 
of c* is: 
a x a, xl + a, x\ -(- 3 b x x x 2 x t = 0. 
The triangle of coordinates is therefore a triangle of inflection, i. e. 
a triangle of which each side contains three points of inflection of 
c*. There being four triangles of inflection, the (2,2)-correspondence 
of the associated points contains four involutory triplets. 
3. We shall now determine the locus of the associated pairs, 
collinear with a given point D. 
In the first place D is a node of the locus; the points D' and 
D" associated with D are the points of intersection of the polar conic 
d 2 of D with the polar line d of D. The locus is therefore a nodal 
biquadratic curve d A . 
The tangents out of D to c* are at the same time tangents to d\ 
for in their tangential points two associated points continually coincide. 
So d' is the conic of Bertini of d\ For an arbitrary nodal c 4 this 
conic contains besides the six points of contact of the tangents out of 
the node, the points of intersection of c 4 with the line connecting the 
two tangential^ points of the node, and the tangents in those “funda¬ 
mental points” to the conic concur in the node'). 
The curve d* is a special curve, because its fundamental points 
coincide with the tangential points D and so that these are at 
the same tim e the points of contact of a double tangent. 
1) See my paper ‘La quartique nodale” (Archires Teyler, t. IX, p. 263). 
