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

CIRCULAR CUBICS.

77

whence AP.AP' = A 02,
and 2AQ = AP + AP'= (b - a) sin2 0 - 2a cos2 0} sec 0.
t
S
P
t
K
A B
When thnv
When the cubic is the inverse of a hyperbola, the loop is the
inverse of the two branches; but when the cubic is the inverse of
an ellipse it follows that if the tangent from A touches the curve
at R, AR = AO and the portion between A and R is the inverse
of the portion beyond R. Also the locus of Q is a circular cubic
of the same species, having a crunode or an acnode according as
the signs of b- a and 2a are the same or different.
127. If the tangents at P and P' intersect at T, the locus of T
is the cuspidal cubic
x {(b- a) x2 + 2by2} = 2by2.
Let (h, k) be the coordinates of T referred to 0 as origin.
Transfer the origin to A and let y = mx be the equation of AP'P.
Then this line must intersect the cubic and the polar conic of T
in the points P, P'; if therefore we substitute mx for y we shall
obtain two quadratic equations which must be identical.
The cubic leads to the equation
x2 (m2 +1)+ {2a +m2(a- b)} + a2 = 0,
and the polar conic to the equation
x2{ 3h - a + 2km + (h - b) m2}
2x {2ah- a2 + (a - b) m} + a2 (h - a) = 0,
whence equating coefficients, we get

ah + k (a-b)m- I (a - b) (h -a) m2 = 0.........(6).