An elementary treatise on cubic and quartic curves, by A. B. Basset.

160

BICIRCULAR QUARTICS.

Let B and C be two real points of inflexion, and let AB, AC
be the tangents at B and C. Then if /3=0, the cubic must
reduce to a = 0, which requires that
n=0, 2m'= vb/c,
Ic = b - 2m'a = b (Xc - va)/c...............(23).
The conditions that the cubic should reduce to a3=0 when
y =0 are that
m = 0, 2n' = vc/b,
lb = c ( b b- a)/b.....................(24).
The third real point of inflexion D must lie on BC, whence
putting a = 0 in (20), the equation of AD is
(f8/3 + vy) a = 21' (b3 + cy).
If D is at infinity, AD must be parallel to BC, whence
t/b = /c = k........................(25).
Using this in (23) and (24) we get
(X- a)(b c ) = 0....................(26).
The solution X= ka must be rejected, because it leads to the
cubic breaking up into a conic and the line at infinity; the other
solution shows that b = c, whence / = v, and the equation of ul
becomes
Xa + A (/ + 7)=0,
which is parallel to BC. These results show that when a circular
cubic has three real points of inflexion, one of which is at
infinity:(i) The tangents at the two other points of inflexion, together
with their chord of contact, form an isosceles triangle of which the
chord of contact is the base.
(ii) The line joining the points of inflexion is parallel to the
asymptote.
242. The following is an example of tangential coordinates.
If through any point 0 on a circular cubic a line be drawn
cutting the cubic in P and Q, and RX be drawn perpendicular to
PQ through the middle point R of PQ, the envelope of RX is a
parabola.