350] 
ON THE CLASSIFICATION OF CUBIC CURVES. 
375 
and thence 
or, what is the same thing, 
\x 2 — fxx (-| — x) — '2/jj (^ — x) 2 = 0, 
(A - fi) x 2 + § fix - \ /j, = 0, 
so that putting for shortness 
/x — A 
= та, (та- denotes the distance of the satellite line from 
the asymptote x — 0) then the equation which determines the distance of the critic centres 
from the asymptote is 
X 2 — ij 'UJX + \ Z7 = 0, 
or we have 
x = J (Зта + V 9та 2 — 8та). 
The condition for a twofold centre is (та- -— 0, which may be disregarded, or else) та- = f, 
or, what is the same thing, 8A + /л = 0. 
69. If x 1} x 2 are the coordinates of the two centres, we have 
and thence 
or, reducing, 
2x 2 = таг (3«! — 1), 
2x 2 = та (Зя?о — 1 ), 
X 2 _ 3#! — 1 
x 2 Sx 2 — 1 ’ 
x, + x 2 — Зх^ = 0, 
a relation connecting the two values x 1 , x 2 ; this equation however only expresses the 
known relation that the two centres are harmonics of each other in regard to the 
twofold centre conic or circle. 
70. The foregoing examination of the form of the envelope shows very readily 
what are the positions of the satellite line which give rise to Plucker’s groups for 
the Hyperbolas A Redundant. 
We have in fact first, 
Hyperbolas A Redundant, no osculating asymptote. 
The satellite line is not parallel to a side of the triangle; and the different 
positions give the following six of Plucker’s groups, viz. 
I. 
IL 
III. 
IV. 
V. 
VI. 
Satellite line cuts three sides produced. 
„ passes through a vertex and cuts opposite side produced. 
„ passes through a vertex and cuts opposite side. 
„ cuts two sides and a side produced, but does not cut the envelope. 
„ touches the envelope. 
„ cuts the envelope. 
Next, 
Hyperbolas A Redundant, one osculating asymptote.