350] 
ON THE CLASSIFICATION OF CUBIC CURVES. 
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I commence by establishing the theory of the classification of cubic curves according 
to the nature of their infinite branches, in what appears to me the scientifically correct 
manner as follows : 
The Seven Head Divisions, Article Nos. 1 to 4. 
1. A line in general, and therefore the line Infinity, meets a cubic curve in three 
points, and these may be 
Three onefold points, 
A twofold point and a onefold (or, as it may also be termed, a one-with-twofold) point, 
A threefold point. 
2. But in the second case the line Infinity 
may be a proper tangent to the curve, 
may pass through a node, 
may pass .through a cusp; 
and in the third case the line Infinity 
may touch the curve at an inflexion, 
may at a node touch one of the two branches, 
may touch the curve at a cusp. 
3. The first case, the three divisions of the second case, and the three divisions 
of the third case, give in all seven divisions, which, as will appear in the sequel, fall 
in with Newton’s classification, and can be named in his language, viz. 
Three onefold points, 
A onefold and a twofold point; 
Infinity a proper tangent, 
Do. through a node, 
Do. through a cusp, 
The Hyperbolas. 
The Parabolic Hyperbolas. 
The Central Hyperbolisms. 
The Parabolic Hyperbolisms. 
A threefold point; 
Infinity a tangent at an inflexion, The Divergent Parabolas. 
Do. Do. at a node, to one branch, The Trident Curve. 
Do. Do. at a cusp, The Cubical Parabola. 
4. As regards the signification of these terms, it may be remarked that the 
Hyperbolas have hyperbolic branches, the Parabolic Hyperbolas, hyperbolic and parabolic- 
branches ; where by a hyperbolic branch is meant one having an asymptote, and by 
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