ON CUBIC CONES AND CURVES. 
mere line which is an acnodal line, or say an acnode; the cone is then of the 
acnodcil kind. Or the two things may happen together, viz. the twin-pair sheet at the 
instant that it unites itself with the single sheet may dwindle into a mere line, which 
is then a cuspidal line, or say a cusp; and the cone is then of the cuspidal kind. 
I remark, as regards the crunodal kind, that the cone may be considered as consisting 
of two portions, one of them corresponding to the single sheet of a complex cone, and 
which I call the quasi-single sheet; the other of them corresponding to the twin-pair 
sheet, and which I call the loop-sheet. 
2. It is to be added that a cubic cone has in general 9 lines of inflexion, or 
say inflexions, but of these 6 are always imaginary; the remaining 3, which are real, 
belong to the single sheet. The plane through any two inflexions meets the cone in 
a line which is also an inflexion. In particular the three real inflexions lie in a plane. 
3. When the cone is acnodal the six imaginary inflexions unite at the acnode; 
and the single sheet has still 3 real inflexions lying in a plane. But if the cone 
is crunodal, then 4 imaginary inflexions and 2 real inflexions unite in the crunode; 
and the cone has 1 real inflexion; there are besides 2 imaginary inflexions, the 3 
inflexions lie in a real plane. Finally, if the cone is cuspidal, then 2 of the real 
inflexions, and the 6 imaginary inflexions unite together in the cusp; the cone has besides 
1 real inflexion, but there are not any imaginary inflexions. 
4. Suppose that the cone is of one of the non-singular kinds; that is, let it be 
simplex or complex. From any line of the cone we may draw four tangent planes to 
the cone—the anharmonic ratio of the four planes is the same whatever may be the line 
on the cone. As regards reality, the following distinction exists, viz. for the complex 
kind of cone, the planes are all real or all imaginary; for the simplex kind they are 
two real and two imaginary. First, as regards the complex kind, if the line be taken on 
the twin-pair sheet, the four tangent planes are all imaginary; but if it be taken on the 
single sheet, then there are two real tangent planes to the single sheet and two real 
tangent planes to the twin-pair sheet, together four real tangent planes. Secondly, as 
regards the simplex kind, there is only the single sheet, and the line being taken on 
it, there are two real tangent planes and no more. 
o. As regards the singular kinds, assuming always that the line on the eone does 
not coincide with the node or the cusp (for when it does there are no tangent 
planes), it may be noticed that for the crunodal kind there are two tangent planes 
which are real or imaginary according as the line lies on the part corresponding to 
the single sheet or on the part corresponding to the twin-pair sheet. For the acnodal 
kind there are two tangent planes which are always real; and for the cuspidal kind 
there is a single tangent plane which is always real. 
6. The foregoing properties of cubic cones apply to the curves which are the 
sections of these cones ; thus a cubic curve is of the simplex, the complex, the crunodal, 
the acnodal, or the cuspidal kind. As regards the last-mentioned three kinds, or singular 
kinds, it is of course to be borne in mind that the crunode, acnode, or cusp, may 
be at infinity; and consequently that all the curves in Newton’s genus 9 (the hyper- 
bolisms of the hyperbola) and the curve in his genus 12 (the trident curve) belong 
to the crunodal kind; the curves in genus 10 (the hyperbolisms of the ellipse) to the 
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