350] 
ON THE CLASSIFICATION OF CUBIC CURVES. 
383 
88. I notice the particular case — - = which corresponds to the direction 
parallel to one of the nodal tangents of the envelope: the harmonic conic is in this 
case the ellipse 
3 x 2 + y- — 2a? — y + xy = 0, 
which, it will be observed, passes through the point (x — 0, y = l) which is one of the 
points of intersection of the twofold centre locus or hyperbola 3a? 2 — 4x — y 2 — — 1 by 
the line x = 0. 
89. For the value — - = V3 corresponding to a direction inclined at an angle = 30° 
to the asymptote, the harmonic conic becomes the parabola (x V3— y) 2 — 2(x — 3/ V3) = 0 : 
this equation may also be written (x — %) 2 + y 2 = l(x + y^/3 — l) 2 , a form which puts in 
evidence the focus and directrix of the parabola. 
90. For a value — ->V3, that is, when the satellite line is inclined to the 
asymptote at an angle < 30°, the harmonic conic is a hyperbola, and ultimately when 
— - = 00, or the satellite line is parallel to the asymptote, the harmonic conic becomes 
f 1 
the pair of lines y (1 — x) = 0. 
91. I have in the figure shown the following forms of the harmonic conic; viz. 
hyperbola, corresponding to inclination < 30° of satellite line to asymptote, 
parabola, to inclination =30°. 
ellipse 
< ( 
ellipse to inclination = < 
ellipse J > 
inclination (=tan _1 2) of a nodal tangent of the envelope, 
and for these forms respectively the successive positions of the satellite line are indicated 
as follows : 
92. For the inclination < 30°, the positions are AGMG'DEA', viz. 
A, at infinity, 
G, touching lower branch of envelope, 
M, between G and G\ 
G', touching upper branch of envelope, 
D, passing through asymptote point D 1} 
E, passing through crunode of envelope, 
Aat infinity.