( 236 ) 
We can determine the order of this curve C; in the following way: 
Construct one of the common tangents 4 of K? and Cs; let 7) be 
the point of contact of t with A», then 7, is a double point of 
the involution. Through 7, still two tangents can be drawn to C3; 
they intersect K? in the branchpoints of the involution; these 
branchpoints are common points of K? and C;. We can conclude 
from the number 6 of the common tangents that there are 12 of 
these branchpoints; therefore C, intersects the conic K? in 12 points. 
Hence C3 is of the sixth order and may be called C°. 
6. The locus of the poles of the tangents of C° in reference to 
K? is the reciprocal polar curve C* of C&; it is of the third order 
and of the sixth class. We imagine once more a point Aj, on C® 
as the pole of chord A, A, of K?; A, and A, determine two points 
of the corresponding conic A? of the pencil which intersects K? 
moreover in A, and A,; A® also intersects C® in the six points 
Ajo A'}3 +» » A’sy. Moreover five other poles will appear on CS 
besides As, the poles of the chords 4,43, A,Ay, Agg, Agoda, AgAy 
By assuming one point on C*, two groups, each of 6, are deter- 
mined on C?, the group A and the group A’. If one of the points 
A is taken arbitrarily no point of group A will coincide with a 
point of group A’. 
7. The following conclusions may be arrived at from the preceding: 
a. If we assume successively the points Ajo, Big, Cia... on C%, 
as many groups of 6 points are formed; each of the points of the 
group can determine the whole group unequivocally, so the points 
A, B, C... form an involution of the sixth order on C?, 
b. The points of intersection A’, B', C'.. . of the conics with 
C® also form an involution of the sixth order. 
c. Each point of group A' corresponds to any point of group 4; 
reciprocally each point of group A corresponds to any point of 
group A’; so both involutions are projective. 
d. The points of coincidence of both involutions determine the 
conics which give the solution of the probiem (3). 
8. The projective involutions on the same bearer are both of 
the sixth order, so they have 12 points of coincidence), These 
points may be indicated more closely in the following manner: 
1) E. Korver, Grundzüge einer rein geometrischen Theorie der algebraischen ebenen 
Curven, (“Elements of a purely geometrical theory of the algebraic plane curves’’.) 
p. 88 etc. 
