td e= AVEN 
ji, = —_ as. — A 2,7), 
In this C, thus oo! quadrangles are described having all O, and O, 
as diagonal points. 
The vertices of these quadrangles evidently form a fundamental 
involution HF, 
Out of 
#,7=( at Hide”, 
Ao CN nd KP 
we find for the diagonals of the quadrangle (2) the equation 
(Aa,* —j) 2,7 = (da, 2 HAf) z°. 
So all quadrangles have in OV, their third diagonal point. 
At the same time it is evident from this that we can build up the © 
above mentioned #, out of pairs of the fundamental #, of which 
each ray through QO, contains two pairs. 
If the two pairs coincide then the ray which bears them is a 
double tangent. 
The pairs on the ray z, = or, we find out of 
fH + (a,*—a,°0°) 4+ fo? = 0 
Thus for a double tangent we have 
(2,7 —a,707) = Ao", 
or 
ao’ + 2fo — a,? — 0. 
So O, ts the point of intersection of four double tangents corre- 
sponding to 
de + rr, — az,’ = 0, 
or, what comes to the same, to 
2 2 a Bie 
a, U, == Zer, x, el i a, Ls a Q, 
The eight points of contact he on a conic. 
For, the polar curve of OU, degenerates into 2, =O and the conic 
2 2 2p 2 nn 
aa, + a,*2,7 — 2c?z,? = 0. 
5. We shall show now that the remaining four double tangents are 
connected with two fundamental mvolutions of pairs which can be 
generated by conics. 
r 
The curve C, can be generated by the projective pencils 
(e, — 4,7) (4, — 4,”,) = ofz,’, 
Q (z, zin a,x) (z, =e a.) = Edge 
