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

THE CONCHOID OF NICOMEDES.

203

produce it to P so that PQ=b; then the locus of P is the
conchoid.
Let AQO = 0, OP = r, then the polar equation of the curve is
r= a cosec 0 +b........................(1),
or ( 2 + y (y - a)2 b2y2.....................(2).
The origin is a double point, the tangents at which are
a2 + (a2 - b2) y2 = 0,
and is therefore a node, a cusp or a conjugate point, according as
a < or = or > b; also the line y =a is an asymptote.
BThe form of te c e is s n in te f; te d d
The form of the curve is shown in the figure; the dotted line
represents the curve when a > b, and the dark line when a < b.
The curve has also a real tacnode at infinity on the asymptote;
for if the origin be transferred to the point A, (2) may be written
in the form
-a2b2- 2aby + y2 {X2 + (y + a)2 - b2} = 0,
which is of the same form as the first equation of ~ 188. If therefore the point 0 is a node, the curve is a trinodal quartic and
belongs to species VII; if on the other hand 0 is a cusp, the
curve belongs to species VIII, and is of the fifth class.
The curve obviously passes through the circular points; hence
a circle which passes through the node cannot intersect the
conchoid in more than four other points. Also if the equation of
the upper portion be r = a cosec + b, that of the lower portion
will be r = a cosec 0 - b.
306. We shall now show how the conchoid can be employed
to trisect an angle.