XIV 6 SYMMETRY OF CUBICS 229

Ex. 1. For what value of h is the pencil of tangents from any point of
ihe cubic (x + y + z) s + 6kxyz = (i) harmonic, (ii) equianharrnonic ?
[(i) & 2 + 6& + 6 = 0, (ii) k = -i.]

Ex. 2. Find the inflexions of the cubic

/= (x + y + z) s +6kxyz = 0.
[The Hessian is

H=(x + y + z)(2yz + 2zx + 2xy-x t -y i -z i ) + 2kxyz = 0.
The inflexions lie on the lines

f-3H=4:(x + y + z)(x + ay + a>''z)(x + ca*y + B)z) = 0.]

Ex. 3. Show that in general two cubics with given inflexional tangents
at three given collinear inflexions touch a given line. Mention any
exceptions.
[If the line passes through an inflexion or the intersection of two of
the given tangents, there is only one cubic]

Ex. 4. Given the tangents at three given collinear inflexions of a
cubic, the locus of the remaining inflexions is two straight lines.
[See Ex. 2.]

Ex. 5. Show that nine cubics can be drawn in general to pass through
three given points, given three inflexional tangents and the fact that the
three corresponding inflexions are collinear.
