[Parallel to the axes of the two parabolas through the foci.]
Ex. 11. The lines joining four concyclic foci of a circular cubic in

pairs are equally inclined to the real asymptote.
Ex. 12. The centroid of any four concyclic foci of a circular cubic is the
foot of the perpendicular from the singular focus on the real asymptote.
[Eliminating y or x from § 3 (ii) and (iii) we get the abscissae or
ordinates of the four foci on (iii).]

Ex. 13. The directrices corresponding to four concyclic foci of a
circular cubic form a pencil of the same cross-ratio as the pencil sub-
tended by the foci at any point of the focal parabola through them.
[The directrices are found by differentiating (iv) with respect to m\
and choosing m so as to make (iv) a line-pair.]

Ex. 14. The centres of the four circles for which a circular cubic is
self-inverse lie on a rectangular hyperbola whose asymptotes are the
real asymptote of the cubic and the perpendicular from the singular
focus.
[The locus of the centre of (iii) is 2(x+p)y + b =0, if t is taken as
a varying parameter.]

Ex. 15. Show that the cubic (i) of § 2 is self-inverse with respect to
x 2 +y 2 = h, and that the corresponding focal parabola is

(y + hf = b(2x + a).
Find an equation giving the other focal parabolas.
Ex. 16. Show that all circular cubics having the same real asymptote
and singular focus, and self-inverse with respect to the same point, have
a focal parabola in common.
