Evidently the two variable points of intersection of conjugate 
conics lie on the line 
2) , : 3 } 
ag (ar, ay at) + (9 Di 1) fw, 0, 
passing through the point /7, having as coordinates (a,, — a 
Each line 
„ 0). 
a,v, + a,v, + ofa, = 0 
bears two pairs of the fundamental involution which can be generated 
by each of the two pencils of conics; for we have 9? — 2690 + 1—= 0. 
For @ = +1 these pairs coincide and we find the double tangents 
ats das, = fr, = 0. 
In a similar way the pencils 
(w, — a,#,) (2, + 4,00) = oft,” 
o (ze, + a,#,) (w, — a,v,) = — fa,? 
determine a fundamental involution which is also generated by the 
rays out of the point 7, (a,,a4,,0), through which at the same time 
the double tangents 
pass. 
The four double tangents form a quadrilateral having O,O0,0, as 
diagonal triangle. 
6. The polar line of (a,, + a,,0) with respect to the conic 
(7, — at) (w, = az) = ofa,’ 
is represented by 
ttr a=, 0. 
From this ensues that the pencils (H,) and (//,) determine two 
involutory (2,2) on these two lines A, and A, Their branchpoints 
are generated by the nodal tangents and the tangents which can 
still be drawn out of O, and O,. 
If we write the equation of C, in the form 
(z,?7 — ar) wv, — (ast, — Cw) «,? = 0, 
it is evident that the lines 
ar =a ca,” 
touch it on rv, = 0. 
In an analogous way the lines 
at, — a Pie 
have their points of contact on #, = 0. 
And now we see directly that these two pairs of rays intersect 
each other on the lines /, and /,, 
at, =a,z,= 0, 
