396 ON THE CLASSIFICATION OF CUBIC CURVES. [350 
Il 
R 
As to the Division into Species: Comparison of Newton and Plucker. 
Article Nos. 121 and 122. 
121. The division into species is obtained without difficulty when the groups are 
once established; in fact it only remains to trace for each given form of V and s the 
series of curves V + ps = 0, as p passes from oo to — oo through the value 0 and the 
critic values which correspond to nodal curves: I have nothing to add to what has 
been done by Plucker, and it is unnecessary to reproduce the investigation. It may be 
remarked that the mere inspection of Plticker’s figures is sufficient to show which of 
his species correspond to the same Newtonian species; the species which do so belong 
to the same Newtonian species in some instances closely resemble each other in form, 
although in others the difference of form is apparent enough: but the Pliickerian species 
which correspond to the same Newtonian species belong for the most part to different 
groups and are thus distinguished from each other by the characters which distinguish 
the groups to which they respectively belong: thus for instance Newton’s Species 1 
(a hyperbola A Redundant) is characterised as consisting of three hyperbolic branches, 
one inscribed, one circumscribed, and one ambigene, with an oval. Such a curve may 
exist with three different positions of the satellite line in regard to the asymptotes, 
viz. the satellite line may cut the three sides produced, or it may pass through a 
vertex, cutting the opposite side produced, or it may cut two sides and the third 
side produced, not cutting the envelope—which are the characters of Pliicker’s groups 
I, II, IV, respectively, and there belongs to the Newtonian species 1, a species out of 
each of these groups, viz. they are I. 1 ; II. 9, and IV. 18. 
122. The correspondence of the Pliickerian Species with those of Newton is shown 
in the following Table. 
Newton’s Genus 1, contains 9 Species, viz. 
1234 5 678 9 
corresponding to Pliicker’s Species 
1. 
II. 
III. 
IV. 
V. 
VI. 
1 
2 
3 
8 
7 
4 
5, 
6 
9 
10 
11 
12 
13, 
14 
15 
18 
17 
19 
16, 20 
21 
22, 
23 
25 
24, 26 
30 
27 
28, 
29 
31 
32, 
33 
Part of Newton’s Genus 4, contains 3 Species, viz.