350] 
ON THE CLASSIFICATION OF CUBIC CUB-YES. 
361 
32. For the Trident Curve the node or threefold point at infinity is real, and 
inasmuch as one of the tangents is the line infinity, the node is a crunode, and the 
other tangent, or asymptote of the curve, is also real. 
33. For the Cubical Parabola, the cusp or threefold point at infinity and the 
tangent at this point are each real. 
Beckoning the hyperbolas as 4, the parabolic hyperbolas as 3, the central hyper- 
bolisms as 2, and the parabolic hyperbolisms, the divergent parabolas, the trident curve, 
and the cubical parabola, each as 1, we have in all 13 divisions. 
The Notion of a Group. Article No. 34. 
34. I remark that the characters as well of the 7 divisions as of the 13 divisions 
have exclusive reference to the form of the asymptotic aggregate V = 0; we have an 
ulterior division depending on the relation of the satellite line to the asymptotic 
aggregate, and which I regard as the proper origin of Plticker s Groups: viz. for a 
given form of the asymptotic aggregate V = 0, and corresponding to each characteristically 
distinct position in relation thereto of the satellite line s = 0, we have a Group. The 
determination of the characteristically distinct positions of the satellite line cannot be 
completely effected a priori; for instance, in the case of the Hyperbolas A redundant, 
the distinctions which immediately present themselves are that the satellite line cuts 
the three sides produced, or two sides and the third side produced, of the triangle 
formed by the asymptotes, or passes through an angle of the triangle, &c.; but these 
are not all the distinctions which have to be made; to determine them, taking the 
satellite line as given, we discuss the series of curves represented by the equation 
V + pz 2 s = 0; for instance (and it is on this that the discussion chiefly turns), we see 
that the parameter p may be so determined that the curve shall have a node, but 
the reality or non-reality of the roots of the equation in p, and therefore the existence 
of a real nodal curve or curves will depend on the position of the satellite line s = 0; 
and it is thus only by the discussion of the group that we arrive at an enumeration 
of the different groups. 
Osculating Asymptotes and other Specialities. Article Nos. 35 to 41. 
35. But Plticker nevertheless, prior to the establishment of his groups, introduces 
certain intermediate divisions as to osculating asymptotes, &c., which have really reference 
to the position of the satellite line ; an osculating asymptote gives rise to a c diameter,’ 
and the diameter is a distinctive character in the Newtonian genera; to explain how 
all this is, I proceed as follows. 
36. The parallel asymptotes of a Central Hyperbolism, the twofold asymptote of a 
Parabolic Hyperbolism and the asymptote of the Trident Curve are singular asymptotes, 
that is, each of them touches the curve at a node or a cusp, and is thus an asymptote 
of three-pointic intersection. Excluding these, and using the term asymptote to denote 
C. Y. 46