16 
§ 7. In the following way we arrive at a (2,2) for which n = 3. 
Let «* be a cubic, p? the polar conic, p the polar straight line of 
P. To P we associate the two points of intersection /, and P, of 
p? with p. The correspondence (2,2) arising in this way, is involu- 
torial, because P and P, may be considered as threefold elements 
in a cubic involution where the points of intersection of PP, with 
a® form a group’), or as the double points of the cyclic projectivity 
defined by this group. The class of this (2,2) is therefore one. 
If P gets on a*, P, and P, coincide with P; P is in this case 
a branch point coinciding with the corresponding double point. If 
on the other hand P gets into a point of inflevion. B, p is a part 
of p*, so that B is a singular point and the stationary tangent is 
a singular straight line. 
If P gets on the Hessian H* of a*, p passes through the double 
point of p’, also lying on the Hessian, and P, coincides with P,, 
so that P is a branch point. The branch curve (V )*° consists therefore 
of a> and M* and these curves are at the same time the locus of 
the-double points. 
When P describes the straight line 7, p? describes a pencil and 
p envelops a conic. In each base point of (p?) there lie therefore 
two points associated to P. As a p* contains moreover the two points 
of intersection with the corresponding p, the straight line ris trans- 
formed into a quadrinodal curve g°. This contains the nine points 
of inflexion of a’, as these correspond to the points in which r cuts 
the stationary tangents. Consequently o° touches a* in the three 
points of intersection of a* with r. 
The derivative of this (2,2) is of the fourth class. For a straight 
line p has four poles and contains therefore the four pairs P,, P, in 
which it is cut by the corresponding four polar conics p?. 
') Kou, Zur Theorie der harmonischen Mittelpunkte. (Sitz. ber. der Akad. 
der Wiss. Wien, Bd. LXXXVIII, S. 424). 
