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

THE TRISECTRIX OF MACLAURIN. 81
The Trisectrix of Maclaurin.
133. A particular case of the nodal cubic is the trisectrix
of Maclaurin, whose equation is
x (x2 + y2) =.a (y2 - 3X2),
which may be constructed as follows.
Let OCO' be a diameter of a circle whose centre is C; through
D, the middle point of OC, draw a straight line perpendicular to
OC; draw OBA cutting this line in B and the circle in A; on
AO produced take a point P such that OP=AB. Then the
locus of P is the required curve.
Let OC= a; AOC =; then
- x = OM = OP cos 0 = AB cos 0,
and
AB = OA - OB = 2a cos 0- la sec 0,
whence
- x = la (4 cos2 0 - 1),
-y- a (4cos2 0- 1) tan 0,
whence eliminating 0, we obtain
x (x2 + y2) = 2a (y2 - 3x2).
By means of ~ 123 it can be shown that the circular points
are points of inflexion, that the third point of inflexion (which
must be real) is also at infinity, and that the line x = 2a is the
inflexional tangent.
The Logocyclic Curve.
134. The logocyclic curve is the inverse of a rectangular
hyperbola with respect to a vertex.
Putting a - b in (3) of ~ 125, the equation of the curve may
be written
x (x2 + y2) + b (X2 y2) =..................... (1),
or rcos 0 + b cos 20 =.....................(2).
B. C. 6