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

THE LEMNISCATE OF BERNOULLI. 169
then QCA = ACP = 0,
= CPY= CQP= r-2- 20.........:.....(1),
x= 0+ 7r- - = 30..................... (2),
r3= ap..........................(3),
p3= a cos 2........................(4).
Equation (4) is the pedal and also the tangential polar
equation of the lemniscate.
Also since the p and r equation of a rectangular hyperbola is
CQ. CP = a2, it follows from (3) that
dr a2
p = r dp = 3 = 1CQ..................... (5).
dp 3=r - 3r
The reciprocal polar is the curve
c = r cos...................... (6)
and the tangential equation is
27a4 ( r2 + q2)2 = {4 - a2 (~2 - 2)}3............... (7),
which shows that the curve is of the sixth class, a result which
follows from the fact that it belongs to species VII.
254. To find the p and r equation referred to a focus.
Let SP=r, HP=r', CP=R; then since C is the middle
point of SH,
r2 + r'2 = 2R2 + a2,
also 2rr' = a2,
whence if r' > r,
R V/2 = r' - r = a/r - r.
By (5)
1 dp _ 1 3R 3 3r
r dr p a2 2r /2 a2 2/2
whence 2 /2a2p = (3a2 - 2r2) r.
255. The angular points of an equilateral triangle move round
the circumference of a circle; prove that the locus of the foci of all
rectangular hyperbolas which circumscribe the triangle and have a
given centre is a lemniscate.