402
ON CUBIC CONES AND CURVES.
[351
of the curves which are the sections of such cone; it appears to me that the true
classification of curves is to divide them according to the cones which give rise to
them; and I consider the present memoir as affording in part the materials for such a
classification of cubic curves, viz. it seems to me that after, in the first instance,
dividing them into the simplex, the complex, the acnodal, the crunodal, and the cuspidal
kinds, the simplex kind should be further divided in the above-mentioned manner; and
that we should establish, lastly, the divisions which relate to the particular mode in
which the cone is to be cut, in order to obtain such and such a curve: in effect,
that the principle of classification, according to the nature of the infinite branches
adopted by Newton in the work above referred to, and by Pliicker in his System der
Analytischen Geometrie (Berlin, 1835), and to which has reference my Memoir On the
Classification of Cubic Curves, [350], should be not the primary, but a secondary principle
of classification. I remark that as regards the division into five kinds, Murdoch, in
his highly interesting work, Newtoni Genesis Curvarum per Umbras, [Lond. 1746], has
not only distinguished the Newtonian species which arise from each of the Divergent
Parabolas, or, what is the same thing, from each of the five kinds of cones (it will
presently appear how the mere inspection of Newton’s figures is sufficient to enable
this), but that he has also shown how the cone must in each case be cut in order to
obtain the particular cubic curve. Murdoch also distinguishes the three forms ampullate,
campaniform and intermediate, of the simplex Divergent Parabola, which correspond to
the simplex quadrilateral, trilateral, and neutral.
I remark also that Pliicker in his work above referred to, Dritter Absclmitt, 98,
has considered the equation of a cubic curve in the form pqr + /as 3 = 0, which is in
fact equivalent to the form (X + Y + Z) 3 + 6kXYZ = 0 used in the sequel, but without
arriving at the results obtained in the present Memoir.
The five kinds of Cubic Cones. Nos. 1 to 7.
1. A cone of any order may comprise two distinct forms of sheet, viz. (1) a twin-
pair sheet, or sheet which meets a concentric sphere in a pair of closed curves such
that each point of the one curve is opposite to a point of the other curve (a cone of
the second order affords an example of such a sheet); the twin-pair sheet may be
considered as consisting of two sheets, each of which may be called a twin sheet:
and (2) a single sheet, viz. a sheet which meets a concentric sphere in a closed curve
such that each point of the curve is opposite to another point of the curve: the
plane affords an example of such a curve. We have five kinds of cubic cones, viz. the
simplex, the complex, the acnodal, the crunodal, and the cuspidal. The cone may
consist of a single sheet; it is then of the simplex kind. Or it may consist of a single
sheet and a twin-pair sheet, it is then of the complex kind: these are the non-singular
kinds. The remaining kinds are singular ones, which are most easily explained by
considering them as limiting forms of the complex kind; the twin-pair sheet may come
to unite itself with the single sheet giving rise to a crunodal line, or say a crunode;
the cone is then of the crunodal kind. Or the twin-pair sheet may dwindle into a
