350] 
ON THE CLASSIFICATION OF CUBIC CURVES. 
393 
and we may consider z = 0 as the equation of the line infinity: writing in the formulae 
z — 1, the critic centres are given as the intersection of the parabola 
with the line 
y 2 + 2 nx + mn = 0, 
vy = \ fin (Sx + m) ; 
and the condition for a two-fold centre is 
4z/ J — Smiifi 2 = 0; 
the equation of the satellite lines corresponding respectively to a two-fold centre is 
4 y 2 — 3 mn — 0 ; 
the lines are real or imaginary according as mn is positive or negative, or (observing 
that the equations x + m = 0, y 2 + nx = 0 give y 2 — mn = 0), according as the line 
x + m = 0 cuts or does not cut the parabola y 2 + nx = 0. Suppose for a moment that 
the line does cut the parabola and that y x is the corresponding value of y, then 
Wj 2 = mn; and the equation 4y 2 — Smn =0 of the satellite lines which correspond 
respectively to the case of a two-folcl centre is y 2 = §y 2 . We have thus y=±y x and 
y = ±\!\y x as special positions of the satellite line fiy + v = 0. In the case where the 
line x + m = 0 touches the parabola y 2 + nx = 0, the value of y x is = 0, and we have 
only the special position y = 0; finally, when the line does not cut the parabola there 
is no special position. 
115. Pliicker’s groups are consequently as follows: 
Parabolic Hyperbolas; ordinary asymptote and five-pointic asymptotic parabola, that 
is the line /iy+v = 0 is not at infinity. 
Asymptote cuts parabola, mn = +. 
XXXVII. Satellite line lies outside the lines y—±y x which belong to the points 
of intersection. 
XXXVIII. Satellite line passes through a point of intersection, that is, coincides 
with one of the lines y—±y x . 
XXXIX. Satellite line lies between the lines y = ±y x and y = ± Vf y. 
XL. Satellite line coincides with one of the lines y = ±^§yi, which give respec 
tively a two-fold centre. 
XLI. Satellite line lies between the lines y = + y x . 
Asymptote touches parabola, viz. m = 0. 
XLIII. Satellite line does not pass through the point of contact. 
XLIV. Satellite line passes through point of contact or its equation is y — 0. 
Asymptote does not cut parabola, viz. mn = —. This gives the single group 
XLII. 
c. v. 
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