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

ON A SPECIAL QUARTIC.

247

with this special class of quartics cannot be generalized for every
trinodal quartic by projection, it is worth while to point out that
the study of a special form may often suggest theorems which,
although incapable of being proved in the general case by projection, are nevertheless true, and may be established by other
methods. We may add that the lemniscate of Gerono, ~ 258, is
one of the simplest quartics having a biflecnode and a tacnode;
whilst the conchoid of Nicomedes, ~ 305, is one of the simplest
forms of a quartic having a tacnode and one other double point,
which may be a node or a cusp.
376. All the projective properties of binodal quartics, in
which the two nodes are of the same kind, may be deduced from
those of bicircular quartics; but if the nodes are different, as in
the case of a quartic having an ordinary node and a flecnode, this
method cannot be employed, but a special investigation is necessary. The projective properties of bicuspidal quartics may be
deduced from those of cartesians.
377. With regard to quartics having a tacnode cusp or an
oscuode; or a rhamphoid cusp or a tacnode with or without
another double point, simple forms may be obtained by taking
the singularity as the origin or projecting it to infinity. And
when there are two singularities, both may be projected to infinity
by performing this operation on the line joining them.
On a Special Q&artic.
378. The general expression for a ternary quartic contains
fifteen terms which may be arranged in five sets of three. The
leading terms of each set are a4; a3/8; a3y; a2,32; a2/y, constant
multipliers being understood.
The first set equated to zero is the equation of a quartic
having twelve points of undulation, four of which are real and the
remaining eight are imaginary. The second set is the quartic
whose properties will now be discussed, whilst the third set represents the same quartic differently situated with respect to the
triangle of reference. The fourth set is the equation of a quartic
having three biflecnodes. The fifth set represents four straight