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ON THE CLASSIFICATION OF CUBIC CURVES. 
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a parabolic branch one not having an asymptote. The hyperbolism of any curve is the 
curve derived from it by altering the ordinate in the ratio of the abscissa to any given 
line 
the expression Central Hyperbolism is used to include Newton’s hyperbolisms of the 
hyperbola and ellipse; and the expression Parabolic Hyperbolism to denote his hyper 
bolism of the parabola. The Divergent Parabolas are curves the branches of which 
ultimately diverge from each other as in the semicubical parabola y 2 = ct?, which is in 
fact one of these curves. The names Trident Curve and Cubical Parabola are not 
generic but specific; it so happens that the genera to which they respectively belong 
contain each only a single species. The names for the several kinds of curves are not 
scientifically-devised ones, but it is convenient to have them such as they are. 
The foregoing seven divisions, uniting in one the Central Hyperbolisms and the 
Parabolic Hyperbolisms, are the six head divisions of Pliicker. 
Asymptotes, Ac. Equations for the Seven Head, Divisions. Article Nos. 5 to 22. 
5. For a Hyperbola there is at each of the points at infinity a tangent, which 
is an asymptote ; and the hyperbola has thus three asymptotes. 
6. For a Parabolic Hyperbola there is at the onefold point at infinity a tangent, 
which is an asymptote. There may be described a conic having with the curve at 
the twofold point at infinity a five-pointic intersection( 1 ). Such conic, as having the 
line infinity for a tangent, is a parabola, and it may be termed the asymptotic 
parabola : the Parabolic Hyperbola has thus an asymptote and an asymptotic parabola. 
7. For a Central Hyperbolism there is at the onefold point at infinity a tangent 
which is an asymptote, and which for distinction may be called the onefold asymptote; 
and at the node or twofold point at infinity there is a pair of tangents which are 
the parallel asymptotes. 
8. For a Parabolic Hyperbolism there is at the onefold point at infinity a 
tangent which is an asymptote, and which may be called the onefold asymptote; and 
at the cusp or twofold point at infinity a twofold tangent which is an asymptote, 
and which may be called the twofold asymptote. 
9. For a Divergent Parabola there is not any asymptote or asymptotic conic ; but 
we may consider an asymptotic cubic, viz. this will be a semicubical parabola (y 2 = a?), 
which is in fact one of the divergent parabolas, the cuspidal divergent parabola, and 
which may be in general so determined as to have at the inflexion or threefold point 
1 I have elsewhere spoken of the conic of five-pointic contact: the expression five-pointic intersection is 
more accurate.