351] 
401 
351. 
ON CUBIC CONES AND CURVES. 
[From the Transactions of the Cambridge Philosophical Society, vol. XL Part i. (1866), 
pp. 129—144. Read April 18, 1864.] 
There is contained in Sir Isaac’s Newton’s Enumeratio Lineament tertii Ordinis 
(1706), under the heading Genesis Curvarum per Umbras, the remarkable theorem that, 
in the same way as the several curves of the second order may be considered as the 
shadows of a circle, that is, the sections of a cone having a circular base, so the 
several curves of the third order, or cubic curves, may be considered as the shadows 
of the five Divergent Parabolas. It was remarked by Chasles, Note xx. to the Aperçu 
Historique (1837), that they may also be considered as the shadows of the five curves 
having a centre (the Newtonian Species 27, 38, 59, 62, 72), and that the theorem may 
be stated as follows, viz. (in the same way that all the curves of the second order 
are the sections of a single kind of cone of the second order, so) all the curves of 
the third order may be considered as the sections of five kinds of cones of the third 
order—and that cutting these in one way we have the five Divergent Parabolas, cutting 
them in another way the five curves with a centre. The nature of these five kinds 
of cones, or, what is the same thing, that of the spherical curves in which they are 
intersected by a concentric sphere, was first pointed out by Mobius in his most 
interesting Memoir, “ Grundformen der Linien dritter Ordnung,” Abh. der K. Sachs. Ges. 
zu Leipzig, 1853. I reproduce in the present memoir the characterisation of these five 
kinds of cones—which I call the simplex, the complex, the acnodal, the crunodal, and 
the cuspidal—and I further develope the geometrical and analytical theory ; in particular 
I arrive at a division of the simplex cones into three subkinds, the simplex trilateral, 
neutral, and quadrilateral. I have throughout spoken of cones rather than of plane 
curves, using however, as far as may be, language which is also applicable to a plane 
curve, thus, instead of lines of inflexion, tangent planes, of a cone, I say inflexions, 
tangents, «See. But the theory of the cone is of course that of the projective j3roperties 
C. V. 51