(570 ) 
intersects 0,0, in a point H, separated harmonically by the line / 
from O, and O,. 
So the pairs of points P,, P, form on C,a fundamental involution 
F., of which each ray through M/ contains two pairs. 
The coincidences of /’, are the points of contact of the tangents 
out of H(y, = by, = — b,, y; = 0). The polar curve of H has as 
equation 
(b, ©, — 6, 4) (@, 2, + 2, by) = 0, 
so it consists of the line 4 and the conic 
et, + #,6,—0. 
The points of intersection of this conic with C,, 
DE 4-28, @, 2, De HL, = 0, 
he ont? = O-and on +27 == be: 
By combining 
0, == EP, 
with the equation of C, we find @, z, + z,’)? =0. So H is the 
point of intersection of two double tangents. 
The points of contact of these double tangents forming two pairs 
of PF, and being generated by the conics 2,4 
position is at hand that #, can also be determined by means of the 
pencil of conics 
se a == 0; the su 
’ 
2 
EZ. 
Indeed, the movable points of intersection of these conies with C, 
lie on the rays 
(1 + 0) 2, + 20% = 0, 
passing through 7, whilst the line A, 
Ot = bi 
is the polar of H with respect to each conic 
D= BE 
Resuming we can say: 
Of a C, with two fleenodal points O, and O, two double tangents 
meet on the connecting line O,O, of the double points. The points 
of contact of the four tangents which it is possible still to draw out 
of their point of intersection to CU, lie on a right line, which contains 
moreover three points of intersection of the tangents r,,s5,,t, out of 
O, with the tangents r,,5,,t, out of O, and the point of intersection 
of the inflectional tangents f, and f, in O, and O,. 
3. From (/,7, 5, 4) =(272 5 é) follows 
(i r, Sy t,) == Gs. ie ty 8,) == (s, ts LF r.) = (t, Sa Ts Ay 
