358 ON THE CLASSIFICATION OF CUBIC CURVES. [350 
a twofold point at infinity it is at least four-pointic; and at a threefold point at 
infinity it is at least six-pointic. 
It follows that the intersections at infinity of the cubic and the asymptotic aggregate 
include the six intersections of the cubic by the line infinity considered as a twofold 
line; and hence the remaining three intersections of the cubic and the asymptotic 
aggregate must lie in a line s = 0 (Plucker’s line S), which I call the satellite line. 
And writing z — 0 for the equation of the line infinity, the equation of the cubic is 
of the form U = V + /iz*s = 0. It is to be observed moreover, that when, as for the 
Hyperbolas and the Cubical Parabola, the intersection at infinity is six-pointic, the line 
•s' = 0 is an arbitrary line; when as for the Parabolic Hyperbolas, and the Divergent 
Parabolas, the intersection is seven-pointic, the line s = 0 meets the cubic in a given 
point at infinity, viz. the twofold or the threefold point at infinity; and when as for 
the Central Hyperbolisms and the Parabolic Hyperbolisms the intersection is eight- 
pointic, the line s = 0 has with the cubic a given twofold intersection at infinity; 
this however merely implies that the line s = 0 passes through the node or cusp at 
infinity, and so imposes only one condition on the line s = 0. Finally, when as in the 
Trident Curve the intersection is nine-pointic, the line s = 0 has with the curve a 
given threefold intersection at infinity ; that is, it coincides with the line infinity, z — 0. 
14. The preceding considerations in regard to the asymptotic aggregate V = 0, lead 
very directly to the best analytical form of the function V, and therefore to that of 
the equation U = V + /.iz 2 s = 0, of the cubic. 
15. For the Hyperbolas; the equations of the asymptotes being p = 0, q = 0, r — 0, 
then we have V=pqr = 0 for the asymptotic aggregate; the satellite line is arbitrary, 
and hence 
Equation of the Hyperbolas is 
pqr + fiz 2 s = 0. 
16. For the Parabolic Hyperbolas. Imagine parallel to the asymptote a line p — 0 
touching the asymptotic parabola; and let the line joining the point of contact with 
the twofold point at infinity have for its equation q=0] the equation of the asymptote is 
p + kz = 0, 
that of the asymptotic parabola is q 2 + Apz — 0, and hence the equation of the asymptotic 
aggregate is (p + kz) (q 2 + \pz) = 0 ; the satellite line passes through the twofold point 
at infinity, or its equation is q + az = 0; hence 
Equation of the Parabolic Hyperbolas is 
(p + kz) (q 2 + Xpz) + /jlz 2 (q + az) = 0. 
17. For the Central Hyperbolisms ; the equation of the onefold asymptote is taken 
to be p = 0, and that of the parallel asymptotes to be q 2 + rcz 2 = 0 ; hence the equation