ON CUBIC CONES AND CURVES.
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acnodal kind; and those of genus 11 (the hyperbolisms of the parabola) and the curve
in genus 14 (the cubical parabola) to the cuspidal kind.
7. In the other genera such of the species as are without a node or a cusp,
belong to the simplex or the complex kind: and the mere inspection of the figure
(Newton’s or Plticker’s) is sufficient to show to which of the two kinds the curve
belongs; in fact, when from any point of the curve there are four real tangents, or
there is else no real tangent, the curve is of the complex kind, but if there are two
and only two real tangents the curve is of the simplex kind. And in the former case
we see whether a branch arises from the single sheet or the twin-pair sheet of the
cone, viz. if from a point on the branch there can be drawn four real tangents to
the curve, the branch arises from the single sheet, but if no real tangent can be
drawn, the branch arises from the twin-pair sheet. And in the crunodal kind we see
which part of the curve arises from the quasi-single sheet, and which part from the
loop sheet.
Ulterior Theory leading to the Subkinds of the Simplex Cones. Nos. 8 to 35.
{Several Subheadings.)
8. But the division of cubic cones may be carried further: we may in fact sub
divide the simplex kind. To show how this is, I consider a cone complex or simplex,
but I attend for the moment only to the single sheet. The cone has on the single
sheet three (real) inflexions lying in a plane. I call this the equator, and I call the
tangent planes at the inflexions simply the tangents; the three tangents do not in
general meet in a line, and they divide space into eight regions; of these there are
two not divided by the equator, and which remain trilateral; the other six regions are
divided by the equator each of them into a trilateral and a quadrilateral region, this
gives six trilateral regions and six quadrilateral regions ; there are thus in all 2 + 6 = 8
trilateral regions (I distinguish them as the 2 and the 6 such regions) and 6 quadri
lateral regions.
9. It is easy to see that for a complex cone the single sheet lies wholly in the
6 trilateral regions, and the twin-pair sheet wholly in the 2 trilateral regions. Imagine
the twin-pair sheet to dwindle into a line and then disappear, that is, let the cone
pass from the complex, through the acnodal, into the simplex kind; the single sheet
of the simplex cone will lie wholly in the 6 trilateral regions; this is one form of
the simplex cone; I call it the simplex trilateral. But there is a different form, viz.
the cone may lie wholly in the 6 quadrilateral regions; this is the simplex quadri
lateral. And there is an intermediate form, viz. the three tangent planes at the
inflexions may meet in a line, the 2 trilateral regions then disappear, and there are
only 12 regions, all of them trilateral, which may be considered as forming two systems,
each of 6 regions, viz. each system consists of three non-contiguous regions on one side
of the equator, and (alternating therewith) three non-contiguous regions on the other
side of the equator: the cone lies wholly in the one 6 regions or in the other 6 regions
and I say that the cone is simplex neutral.
