350] 
ON THE CLASSIFICATION OF CUBIC CURVES. 
387 
we have then 
0=MLz£) 
and the values of z are given by the equation 
A (1 — z) 2 — \z(l—z) — 2vz 2 = 0, 
2 (A - v) z 2 - 3\z + A = 0, 
A 
that is 
which, putting 
— 1 — zr, 01’ zr = 
v — A 
the asymptote 2 = 0), becomes 
A — v 
{zr is the distance of the satellite line from 
have 
2 z 2 — 3 zrz 4- zr = 0, 
Z—\ {3ct + ^ ZT (9ZT — 8)}. 
The condition for a twofold centre is (zr = 0 which may be disregarded, or else) 
9-57 — 8 = 0; 
or, what is the same thing, A + 8^ = 0. 
100. If z lt z 2 are the coordinates of the two critic centres, then we have 
2Zi — zr (3z 1 — 1), 
2Z 2 = ZT (3^2 — 1), 
and thence 
or, reducing, 
or in terms of the «-coordinates 
z 2 _Sz 1 — l 
i?“3ÏT-1 ; 
3^02 — z 1 — z 2 = 0, 
3«!«2 — 2 («! + «2) + 1 = 0, 
which equation however merely expresses that the two centres are harmonics to each 
other in regard to the twofold centre conic 3« 2 — 4« — y 2 + 1 = 0. It is right to remark 
that the formulae, although referring to a different system of coordinates, are absolutely 
identical with those given Nos. 67—69, writing therein v for ¿t, and z for x. 
101. An inspection of the form of the envelope shows what are the positions of 
the satellite line which gives rise to Pliicker’s Groups for the Hyperbolas A Defective. 
We have in fact, 
Hyperbolas A Defective, asymptote not osculating. 
49—2