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

THE ELLIPTIC LIMACON.

191

which is the equation for determining the vectorial angles of the
four tangents drawn from (h, k) to the curve. This may be
written in the form
(a2 - bh) sin2 0 + bk sin2 0 + (a2 + b2 + bh) cos2 0
= a {k sin 8 + (2b + h) cos }...(1:).
Let PFP' be the chord; then if 0 be the vectorial angle
of P, 7r+0 must be that of Q; whence, if (h, k) be the point
of intersection of the tangents at P and P', (13) must be satisfied by 0 and 7r + 6. This requires that both sides of (13) should
vanish, whence eliminating 0 between the two equations formed
by equating both sides of (13) to zero, we obtain
(a2 -bh) (h + 2b)2 + kl (a2 - 3b - bh)= 0.
Transferring the origin to the point - 2b, this becomes
bx (2 + y2) = (a + 2b) + (a2- b22 2) y2.........(14),
which is the inverse of a conic with respect to its vertex. When
a= b, the limacon becomes a cardioid and (14) reduces to the
circle
x2 + y2 = 3bx,
the centre of which is the triple focus.
288. The form of (12) shows that the radius of the focal
circle is equal to ~a, and that the distance of its centre from
the nodal focus is equal to lb. Since the radius 8 of the fixed
circle vanishes on account of the limacon being the inverse of
a conic, the theorem of ~ 206 becomes:If from the nodal focus F a line be drawn to meet the curve
in P, and if FQ be drawn to meet the normal at P in Q, such
that the angle FPQ = PFQ, the locus of Q is the focal circle.
Also if F1 be the external focus, the theorem becomes:If from the external focus F1 a chord F1PQ be drawn, the
normals at P and Q intersect on the focal circle corresponding
to F1.
289. To finld the p and r equation of the curve.
We have, by the ordinary formulae,
p=rsin b, bsin =rcot b,
r2 b2 sin2 0
whence -= 1 +.p2.