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

198

SPECIAL QUARTICS.

at P, FY perpendicular to PT, AFY= X. Then the following
results can be easily proved:
FP = FA. FY2....................... (2),
FY = 2a cos3.....................(3).
P
iA =s T
A C E F
The first equation is the p and r equation of the curve; whilst
the second is the pedal with respect to the cusp, or it may be
regarded as the tangential polar equation of the curve.
Another form of the tangential polar equation is sometimes
useful. Transfer the origin to the triple focus E, then since
FE = a, we obtain
p'=FY- a cos
= 1 a (4 cos3 X - cos X)
= a cos................................. (4).
Equations (3) and (4) are the tangential polar equations of
the curve referred to the cuspidal focus F and the triple focus E
respectively.
From (3) or directly, the tangential equation in Boothian
coordinates is
27a2 ( t2 + 2) = 2 (2 + a~)3.................. (5).
Equation (5) shows that the cubic
27 (2 + y2) c = 2 (2c + )3
is the reciprocal polar of a cardioid; and if the origin be transferred to the point x= 4c, the cubic becomes
2x3 = 9c (2- 3y2),
which has a crunode at the origin, and therefore two of its three
points of inflexion are imaginary. Hence a cardioid has one real
double tangent, one real cusp at the origin, and two imaginary