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

THE LEMNISCATE OF GERONO.

171

Let CQ = r', then by ~ 253,
p=3 r', and OK =r',
whence if (x, y) be coordinates of 0,
x = r' cos 0 - - r' cos 30 =3 r' cos3 0,
Y^ = i r' s in 3 = r' sin 3 0,
whence x] + yf = (2r)3;
also - = (2 r') (cos4 0 - sin4 0)
= (2r') cos 20,
whence (x3 + yt)(x3 - y) = r2 cos 20
= 4a2
which is the equation of the evolute.
257. The lemniscate of Bernoulli, being the pedal of a
rectangular hyperbola with respect to its centre, belongs to the
class of curves included in the equation
(x2 + y2)2 = a2x2 + b2y,
which is the pedal and also the inverse of a central conic with
respect to its centre. These curves are trinodal quartics having a
pair of ordinary nodes at the circular points, and a biflecnode at
the origin which will be complex or real according as the conic is
an ellipse or hyperbola. They are also included in the equation
x + ty = c sec (: + a)), and are one of the few classes of curves
whose potentials can be completely investigated. The two points
which are the inverses of the foci of the conic, and also the
biflecnode at the origin, are double foci. It also follows that if
x and y are tangential coordinates, the preceding equation represents the first negative pedal of a central conic with respect to
its centre; hence Plicker's numbers for the pedal are n= 6,
= 4,'=6, m=4,, =3, =0.
The Lemniscate of Gerono.
258. This curve has been sometimes confounded with Bernoulli's lemniscate, owing to its form being that of a figure of
eight. It may be constructed as follows. Let P be any point
on a circle whose centre is C and radius a; draw PM perpendicular