382 
ON THE CLASSIFICATION OF CUBIC CURVES. 
84. The asymptotes of the harmonic conic meet at the centre; and they again 
cut the twofold centre conic in two points, the intersections of the last-mentioned 
conic with the line 
(k + fii) \(x + yi) + (— k + fii) | (x — yi) — 2fiiz — 0 
that is 
or, what is the same thing, 
fix + ay — 2/xz = 0, 
3 ¡ix + icy — 2/x = 0. 
85. I remark that the equation of the asymptotes of the harmonic conic is 
(Sy 2 — K?) [/x (Sx 2 + y 2 — 2x) + 2icy (1 — x)] + /x(fx 2 + k 2 ) = 0, 
and that the theorem for the construction of the asymptotes depends on the identical 
equation 
(3y? — k 2 ) \/x (Sx 2 + y 2 — 2x) + 2icy (1 — x)] + /x (/x 2 + k 2 ) + S/x (/x 2 + k 2 ) (Sx 2 - y 2 — 4« + 1) 
= — 2 [— (S/x 2 — k 2 ) x + 2/xtc y + fx 2 + /r] (3¡xx + icy — 2/x), 
which is easily verified ; and where 
— (S/x 2 — k 2 )x+ 2fXK y + /x 2 + k 2 = 0 
is the equation of the tangent of the twofold centre conic Sx 2 — y 2 — 4x +1 = 0 at the 
centre of the harmonic conic. 
86. On account of the symmetry in regard to the axis of x, it will be sufficient 
to attend to the series of curves corresponding to a direction y — — — x of the satellite 
line, for which the ratio — - has a given sign; and the inclination of the satellite 
A 4, 
line to the asymptote will pass from 90° to 0° according as the value of the ratio 
— - passes from 0 to oo. 
87. First, if — - is =0, that is, if the satellite line be perpendicular to the 
asymptote, then the harmonic conic is the ellipse 
Sx 2 + y 2 — 2x = 0, 
or, as it may also be written, 
9(x-±) 2 + Sy 2 =l. 
As — - increases from 0 to ^3, that is, as the inclination of the satellite line 
diminishes from 90° to 30°, the harmonic conic becomes an ellipse of greater and 
greater excentricity, and ultimately a parabola.