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ON THE CLASSIFICATION OF CUBIC CURVES. 
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the inflexion, a line which cuts harmonically the chords through the inflexion, and which 
when the inflexion is at infinity becomes a diameter. The remarks previously made as 
to osculating asymptotes apply therefore to diameters, viz. the Hyperbolas may have no 
diameter, a single diameter, or three diameters, &c. 
40. Newton speaks also of the “ centre ” of a cubic curve; viz. there may be a 
point on the curve such that for any line through this point the two radius vectors 
are equal and opposite to each other, or that the sum r + r' of the two radius vectors 
is = 0. The centre is in fact a point of inflexion which has for its polar the line 
infinity. The curves which may have a centre are the Hyperbolas (redundant or defective), 
the Central Hyperbolisms (of the hyperbola or ellipse) and the Cubical Parabola. For 
the hyperbolas, the three asymptotes and the satellite line must meet in a point of 
the curve, which point is then the centre; for the central hyperbolisms the onefold 
asymptote and the satellite line must meet in a point of the curve, which point is 
then a centre; and for the cubical parabola no condition is required, but the inflexion 
is a centre. I remark here, in passing, that the notion of a centre as just explained 
has no place in Plucker’s Classification, and that the two Newtonian species 58 and 59 
(hyperbolisms of the hyperbola) and the two Newtonian species 61 and 62 (hyperbolisms 
of the ellipse) which differ, the two of a pair from each other, according as there is 
no centre or a single centre, form each pair a single species with Plticker; viz. they 
are 198 and 201 respectively. 
41. It has been already remarked that the three asymptotes of a Hyperbola may 
meet in a point. As to this it is to be noticed that from any point we may draw 
six tangents to a cubic, the points of contact lie on a conic, the conic polar of the 
point: if, however the point lie on the Hessian of the cubic, then the conic breaks 
up into a pair of lines, each of which is a tangent to the Pippian; the two lines 
meet in a point of the Hessian, which point forms with the first mentioned point a 
pair of conjugate poles of the cubic( J ). 
Conversely, any tangent of the Pippian meets the cubic in three points, the 
tangents at which meet in a point of the Hessian; and from this point we may draw 
to the cubic three other tangents the points of contact of which lie on a line which 
is also a tangent of the Pippian, and the two tangents of the Pippian meet in a 
point of the Hessian; the two points of the Hessian being conjugate poles of the 
cubic. In particular, if the line infinity is a tangent of the Pippian, then the three 
asymptotes meet in a point of the Hessian, and the three tangents from this point to 
the cubic touch the cubic in three points lying on a line which is a tangent of the 
Pippian, and which meets the line infinity in a point forming with the first mentioned 
point a pair of conjugate poles of the cubic. 
I proceed now to explain the classification of Newton so far as relates to the 
division into genera, and the classification of Pliicker so far as relates to the divisions 
immediately superior to the groups. 
1 See as to this theory my Memoir on Curves of the Third Order. Phil. Trans, p. 147 (1856), [145]. 
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