350] 
ON THE CLASSIFICATION OF CUBIC CURVES. 
359 
of the asymptotic aggregate is p (cf + nz 2 ) = 0, the satellite line passes through the 
twofold point at infinity, its equation is q + az = 0; hence 
Equation of the Central Hyperbolisms is 
p (q 2 4- kz 2 ) + pz 2 (q + crz) = 0. 
18. For the Parabolic Hyperbolisms: the only difference is that instead of the 
parallel asymptotes q 2 + kz 2 = 0 we have the twofold asymptote q 2 = 0 ; hence 
Equation of the Parabolic Hyperbolisms is 
pq 2 + pz 2 (q + az) = 0. 
19. For the Divergent Parabolas: the asymptotic aggregate is a semicubical 
parabola; let q = 0 be the equation of the cuspidal tangent, p = 0 the equation of the 
line joining the cusp with the inflexion at infinity, then the equation is p 3 + \q 2 z = 0. 
The satellite line passes through the threefold point at infinity, its equation is p+az—O, 
hence 
Equation of the Divergent Parabolas is 
p 3 + \q 2 z + pz 2 (p + az) = 0. 
20. For the Trident Curve: let p = 0 be the equation of the asymptote, q = 0 
that of the tangent to the asymptotic parabola at the point not at infinity where it 
is met by the asymptote, then the equation of the parabola is p 2 + \qz = 0, and that 
of the asymptotic aggregate is p (p 2 + \qz) = 0; the satellite line is the line infinity, 
z — 0; hence 
Equation of the Trident Curve is 
p (p 2 + Xqz) + pz 3 = 0. 
21. For the Cubical Parabola: let p = 0 be the equation of the line joining the 
inflexion with the cusp at infinity, then the asymptotic aggregate is this line taken 
as a threefold line, or the equation is p 3 = 0; the satellite line is arbitrary; hence 
Equation of the Cubical Parabola is 
p 3 + pz 2 s — 0. 
22. It is convenient to notice here that for the Hyperbolas the line s = 0 is 
determined as follows, viz. the line infinity meets the curve in three points, and the 
tangents at these points (the asymptotes) again meet the curve in three points lying 
in a line which is the line in question; in other words, the line s = 0 is (in the sense 
in which I have elsewhere used the term) the satellite line of infinity. For the other 
kinds of cubic curves, the line s = 0 is not, in the sense just referred to, the satellite 
line of infinity: but in the present Memoir I shall in every case call the line, s = 0, 
the satellite line.