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

SINGULARITIES AT INFINITY.

125

points at infinity belongs to the theory of curves of a higher
degree than the fourth. Quartic curves having nodes or cusps at
the circular points will be considered in the next chapter, whilst
the investigation of the equations of quartics having imaginary
points of inflexion or undulation at the circular points may be
left to the reader.
189. We have shown in ~~ 79 and 80 that if a curve of class
m has a pair of nodes at the circular points, and in addition has 8
nodes and K cusps, the curve has two double foci and m- + 28 + 3c -4
single foci, which may however for certain values of the constants coalesce into one or more multiple foci. If however the
curve has a pair of flecnodes at the circular points, the point of
intersection of the two inflexional tangents will be a triple focus,
and consequently the curve will have m + 28 + 3c - 5 single foci,
one triple and one double focus. And if the circular points are
biflecnodes, the curve will have m + 28 + 3K - 6 single foci and two
triple foci. We shall hereafter show that the Cassinian, for which
mn = 8, 8 =, K = 0, has a pair of triple foci and a pair of single
foci; whilst the lemniscate, which is a particular case of the
Cassinian, for which m= 6, 8 = 1, K = 0, has a pair of triple foci
and a double focus at the real biflecnode which is formed by the
union of the two single foci of the Cassinian.
Binodal Quartics*.
190. The general equation of a binodal quartic whose nodes
are B and C is
a3u + X2/2y2 + /2y2a2 + v2a2fl2 + 2a/qyv = 0.........(1),
where u = L2a + M/3 + N7,
v = la + m/n + n,.
* The theory of anautotomic quartics has been considered by Zeuthen in a
series of memoirs published in the Mathematische Annalen, where a variety of
papers by Brill, Klein and other German mathematicians bearing on the subject
will be found. Uninodal quartics have been discussed by W. R. W. Roberts,
Proc. Lond. Math. Soc. Vol. xxv. pp. 151-172; and unicuspidal quartics by H. W.
Richmond, Quart. Journ. Vol. xxvII. p. 5. Reference may also be made to The
Forms of Plane Quartic Curves by Miss Gentry, published by Robert Drummond of
New York; to the Index of Papers, Proc. Lond. Math. Soc. Vol. xxx.; to Prof.
Cayley's Collected Papers; and to the papers of H. M. Jeffrey in the Quarterly
Journal.