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

188

SPECIAL QUARTICS.

281. Putting FIP = r1, PFF= 0, i(a2 - b2)/b =f= FF1, the
polar equation of the curve referred to F1 is
2 -ri (b + a2 cos 0)/b 4 f2 = 0...............(3).
Whence if F1PQ be any chord, the locus of the middle point of
PQ is the hyperbolic limaton
r - (b2 + a2 cos )/b.....................(4),
also F1P. FQ = FF2.......................(5).
Equation (5) shows (i) that the curve is its own inverse with
respect to the external focus F, which is therefore a centre of
inversion; (ii) that the triangles F1QF and FF1P are similar; (iii)
that the circle circumscribing FPQ touches FF1 at F. Also from
the properties of inverse curves, the angles TPQ and TQP, made
by the tangents at P and Q with F1P, are equal.
282. Let the tangent at P meet FF1 in L; let FPL=P,
PLF =; then
tan 4 = (a - b cos 0)/b sin 0,
a cos - b cos 20
tan b sin 20 - a sin 0
The form of the curve shows that at the point of contact of
the double tangent, b = rr- 0, whence
cos = a/b r= a.....................(6).
Accordingly the points of contact of the double tangent will
be real if a < 2b.
Making r a minimum we obtain
co 0= (a2 + 2b2)/3ab...................(7),
which determines the two points of inflexion. In order that they
may be real it is necessary that 2b > a > b. When a = 2b, the
vertex A is a point of undulation.
283. The Cartesian equation of a lima9on is
(x2 + y2 + bx)2 = a2 (2 + y2),
which shows that the origin is a conjugate point or a crunode
according as the limacon is elliptic or hyperbolic; also since the
curve is of the ninth species, its reciprocal polar is another quartic
of the same species. When 2b > a > b, the reciprocal curve has
two real cusps, one crunode, a real double tangent touching the