ON QUARTIC CURVES.
[From the Philosophical Magazine, vol. xxix. (1865), pp. 105—108.]
The expression ‘an oval’ is used, in regard to the plane, to denote a closed curve
without nodes or cusps; and, in regard to the sphere, it is assumed moreover that the
oval is a curve which is not its own opposite, and does not meet the opposite curve ( J )—
that is, that the oval is one of a pair of non-intersecting twin ovals. I say that
every spherical curve of the fourth order (or spherical quartic) without nodes or cusps
may be considered as composed of an oval or ovals lying wholly in one hemisphere
(that is, not cutting or touching the bounding circle of the hemisphere), and of the
opposite oval or ovals lying wholly in the opposite hemisphere; or, disregarding the
opposite curves, that it consists of an oval or ovals lying wholly in one hemisphere.
And this being so, the quartic cone having its vertex at the centre of the sphere is
met by a plane parallel to that of the bounding circle in a plane quartic curve con
sisting of an oval or ovals; and thence every plane quartic is either a finite curve
consisting of an oval or ovals, or else the projection of such a curve.
Considering first the case of the plane, a line in general meets the oval in an even
number of points (the number may of course be = 0); hence as the point of contact
of a tangent reckons for two points, the tangent at any point of the oval again intersects
the oval in an even number of points (this number may of course be = 0). The number
of points of intersection by the tangent (the point of contact being always excluded) is
either evenly even, and the point is then situate on a convex portion of the oval; or
it is oddly even, and the point is then situate on a concave portion of the oval.
Now imagine that the oval is (or is part of) a quartic curve; the number of points
1 The notions of opposite curves, &c. are fully developed in the excellent Memoir of Möbius, “Ueber die
Grundformen der Linien der dritter Ordnung,” Abh. der K. Sächs. Ges. zu Leipzig, vol. i. (1852), to which I
have elsewhere frequently referred.
