350] 
ON THE CLASSIFICATION OF CUBIC CURVES. 
357 
at infinity a seven-pointic intersection. For the asymptotic cubic, in order that it may 
have a cusp, must satisfy two conditions; it may therefore be made to satisfy seven 
more conditions, or to have a seven-pointic intersection; and then the original curve 
having an inflexion or threefold point at infinity, the asymptotic cubic will ipso facto 
have the same point as an inflexion, or threefold point at infinity, and be thus a 
cuspidal divergent parabola. 
10. For the Trident Curve, we have at the node or threefold point at infinity, 
viz. to the branch which is not touched by the line infinity, a tangent which is an 
asymptote: this cuts at the node the other branch of the curve, and it is therefore 
an asymptote of three-pointic intersection. We may describe a conic having at the 
node a five-pointic intersection with the other branch of the curve; such conic as 
touching the line infinity is a parabola, and it may be called the asymptotic parabola; 
since the parabola cuts at the node the first-mentioned branch of the curve, viz. the 
branch not touched by the line infinity, the parabola is in fact a parabola of six- 
pointic intersection. The Trident Curve has thus an asymptote and an asymptotic 
parabola of six-pointic intersection. 
11. For the Cubical Parabola there is not any asymptote or asymptotic conic: 
the curve qua curve having a cusp (viz. the cusp or threefold point at infinity) has 
a single inflexion; and the line joining the cusp with the inflexion, regarded as a 
threefold line, has with the curve a six-pointic intersection at infinity, and may be 
considered as an asymptotic cubic. 
12. We have in every case a cubic curve V= 0 having with the original curve 
an intersection at infinity which is at least six-pointic, and which I call the asymptotic 
aggregate : viz. the asymptotic aggregate is 
For the Hyperbolas; the three asymptotes, intersection six-pointic. 
For the Parabolic Hyperbolas; the asymptote and the asymptotic parabola, inter 
section seven-pointic. 
For the Central Hyperbolisms; the onefold asymptote and the parallel asymptotes, 
intersection eight-pointic. 
For the Parabolic Hyperbolisms; the onefold asymptote and the twofold asymptote 
regarded as a twofold line; intersection eight-pointic. 
For the Divergent Parabolas; the asymptotic semicubical parabola, intersection 
seven-pointic. 
For the Trident Curve; the asymptote and the asymptotic parabola, intersection 
nine-pointic. 
For the Cubical Parabola; the line joining the cusp at infinity with the inflexion, 
regarded as a threefold line, intersection six-pointic. 
13. I have said that the intersection at infinity is at least six-pointic; but more 
than this, the intersection at any onefold point at infinity is at least two-pointic; at