380 
ON THE CLASSIFICATION OF CUBIC CURVES. 
[350 
which is 
oi- 
or finally 
Sx 2 — \>x — y 2 — — 1, 
3 (x — |) 2 — y 2 = 
9 0-f) 2 -3i/ 2 = l, 
which is a hyperbola having its centre at the harmonic point x = f, y = 0; having 
¿e = i., a; = 1 for the extremities of the transverse axis, and such that the asymptotes 
are inclined to the axis of x at an angle = 60°; this curve is also shown in the figure. 
79. Similarly making the transformation in the equation of the one-with-twofold 
centre locus written under the form — (— x + y + z) (x — y + z) (x + y — z) + xyz = 0, this 
becomes 
— (— yi + z) (yi + z) (x — z) + \(x + yi) (cc — yi) z = 0, 
that is 
which is 
— 4 (y 2 + z 2 ) (x — z) + (x 2 + y 2 ) z — 0, 
— 4¿ 2 (x — z) + x 2 z + y 2 (5z — 4<x) = 0, 
or, what is the same thing, 
0 (x — 2z) 2 + y 2 (5z — 4x) = 0, 
or, putting for z its value = 1 — x, this is 
that is 
(1 — x) (Sx — 2) 2 + y 2 (5 — 9x) = 0, 
y 
r2 _ _ (x-l)(Sx-2) 2 
9x— 5 
which is the equation of the one-with-twofold centre locus. 
80. The curve is symmetrical in regard to the axis of x. And moreover 
x < |, y is impossible, 
x = |, y 2 — oo , or the line x = | is an asymptote, 
x = |, y 2 = 0, which is a crunode, 
and 
x = 1, y 2 = 0, 
x > 1, y is impossible. 
The equation of the tangents at the crunode are y 2 = S (x — f), or the tangents are 
inclined to the axis of x at angles = 60°. The curve consists, as shown in the figure, 
of a single branch cutting itself in the crunode, and tending on each side towards the 
asymptote.