360 
ON THE CLASSIFICATION OF CUBIC CURVES. 
[350 
The Thirteen Divisions. Article Nos. 23 to 33. 
23. The characters of the foregoing seven divisions are irrespective of reality; and 
before going further it may be remarked, that as to the Hyperbolas and the Parabolic 
Hyperbolas a subdivision also irrespective of reality may be made as follows. 
24. For a Hyperbola, the three asymptotes may not meet in a point, or they may 
meet in a point. For shortness I say that in the former case we have a Hyperbola A, 
in the latter case a Hyperbola ©. I consider more particularly (post, No. 41) the 
special case of a Hyperbola ©. 
25. For a Parabolic Hyperbola, the asymptote may meet the asymptotic parabola 
in two onefold points; or in a twofold point. 
26. I come now to the divisions which depend on reality: it is assumed that the 
curve is real. 
27. For the Hyperbola the three points at infinity may be all real or else one real, 
two imaginary. In the former case, the asymptotes are all real, and we have the 
redundant hyperbola; in the latter case the real point at infinity gives rise to a real 
asymptote, the imaginary points to imaginary asymptotes: we have in this case the 
defective hyperbola. It is to be noticed that the imaginary asymptotes meet in a real 
point, called the asymptote-point; and that such point, if we regard it as an indefinitely 
small ellipse given as to the position and ratio of its axes, determines the imaginary 
asymptotes. Combining the division with the A, ©, we have four subdivisions of the 
Hyperbola. 
28. For a Hyperbola A redundant the three asymptotes form a triangle, and for 
a Hyperbola © redundant they meet in a point. For a Hyperbola A defective, the 
asymptote-point does not lie on the real asymptote; for a Hyperbola © defective it 
does lie on the real asymptote. 
29. For a Parabolic Hyperbola: the onefold point and the twofold point at infinity 
are of necessity real, as are also the asymptote and the asymptotic parabola. If the 
asymptote meets the asymptotic parabola in two onefold points, these may be both real 
or both imaginary: if it meets it in a twofold point, this is real. We have thus 
three subdivisions of the Parabolic Hyperbola. For the Central Hyperbolism, the 
onefold point, and the node or twofold point at infinity, are both real; the asymptote 
is also real. But the node may be a crunode or an acnode ; that is, the tangents at 
the node, or parallel asymptotes, may be both real, or both imaginary: we have thus 
two subdivisions, viz. the Hyperbolism of the hyperbola, and the Hyperbolism of the 
ellipse. 
30. For the Parabolic Hyperbolism, the onefold point and the cusp or twofold 
point at infinity, and also the onefold asymptote and the twofold asymptote are all real. 
31. For the Divergent Parabola, the inflexion or threefold point at infinity is real.