14 
common with the 6? through A. Of tbe locus «a of the points A’ a 
5? contains four points besides the base points 4; they are defined 
by the points of intersection of 6° with a’. On each of the two 
tangents a through Bj, A’ coincides with Bj; hence « has double 
points in Bx. 
Consequently the curve in question is an a’. As it corresponds 
point for point to a’? and is therefore rational, it must have six 
more double points. There are therefore six points A’ each corre- 
sponding to two points A; the 6’? through such a point A’ cuts a’ 
in the two points A which it has in common with the polar line of A’. 
The straight line bj is transformed by (2,2) into a p* with triple 
point 8e When P moves along 6; the polar line p continues to 
pass through 87, so that always one of the points P,, P, associated 
to P, coincides with 8x. If also the second point is to coincide with 
B, p must touch the 6? through Pat 87. Now any straight line p through 
Br touches one 6°; if we associate the points Q,, Q, which this 6? 
defines on bz, to the pole P of p, there arises a correspondence 
(1,2) between P and Q. Hence Q coincides three times with P; 
but then the curve 8: into which bj, is transformed, has a threefold 
point in By and is therefore a rational B“. *) 
§ 4. We shall now try to find the locus of the double points 
P,=P,. It has in the first place threefold points in By. On each 
6? there lie besides the base points four more points of the curve 
in question, namely the double points of the (2,2) in which the 
points of 6 are arranged. Consequently it is a d*. As it corresponds 
point for point to the branch curve (V)° it is just as the latter of 
the genus six; hence it must have three more double points. These 
we find in the double points of the three pairs of lines belonging to 6’. 
The bearers of the double point P, = P, envelop a curve of the 
sixth class (§ 1) of the same genus as the branch curve, hence with 
four double tangents; these we find in the straight lines bz. 
For the points where 4; is touched by two of the conics 6°, 
correspond as double points to the branch point Bx. 
1) On 6; there lie 2 points that are associated in the (2,2) to each other and 
at the same time to Bp, and which therefore together with that point form a 
polar triangle of a? The b? containing them is consequently circumscribed to oc! 
polar triangles so that on it the (2,2) has been transformed into a cubic involu- 
tion In this involution each base point B is associated to the points of intersection 
of b® with the polar line of B. : 
If we define the pencil (62) by two conics, each circumscribed to a polar triangle 
of a? each 6? bears a cubic involution and the whole correspondence (2,2) is 
transformed into a system of oo ! involutorial triplets. 
