An elementary treatise on cubic and quartic curves, by A. B. Basset.

254 ADDENDA AND CORRIGENDA.
former line is the Cayleyan; whence eliminating (h, k) between (1) and
(2), the tangential equation of the Cayleyan will be found to be
a2 (-2 _ 2) = + 2,
which represents a conic.
III. In connection with trinodal quartics, the following theorem
due to Ferrers* may be noticed. His proof is instructive since it
illustrates a method by which properties of trinodal quartics may be
derived from conics. The theorem is:The six stationary tangents of a trinodal quartic touch a conic.
Let the equation of the trinodal quartic be
X2y2 + la + va22 + 2ayy (la + m ny) =......... (1),
and that of any tangent be
da + + - o-............................(2).
If one of the coordinates, say y, be eliminated, the condition that
(2) should be a stationary tangent is that three of the roots of the
resulting equation in a//3 should be equal. If therefore we write i/a,
1//, 1/y for a, /1, y the conditions are the same as that the conics
Xaa2 ~ + Vy2 + 21/y2 + 2mya + 2na = 0..............(3)
and tf7y + -rya + afl = 0.............. (4)
should have a contact of the second order with one another. Writing
these in the form S = 0, S' = 0, it follows that if k be determined so
that the discriminant of S + kS' — 0 vanishes, the last equation will
represent the three pairs of straight lines which can be drawn through
the points of intersection of S and S'. By (4) of ~ 2, it follows that if
A, A' be the discriminants of S and S' the discriminant of S+ kS' when
equated to zero is
A 3 + 3~k2 + 3 'k + A'= 0.....................(5),
where
30 = 2 (mn - IX) + 2 (nl - mp) 7 + 2 (Im - nv),
30= - _X - r_ 2 - v-2 + 215C + 2m5$ + 2n7q,
A' = 2.~.
The conditions that the conics S and S' should have a contact of
the second order with one another are that the three roots of (5) should
be equal; which by (13) of ~ 7 are that
A 0 ~'....................... (6).: 0-' -- a **..............................

* Quart. Journ., vol. xvIIm. p. 73.