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

96

SPECIAL CUBICS.

The Witch of Agnesi.
153. Let AB and CD be two perpendicular diameters of a
circle. Through A draw ANQ, cutting the circle in Q and the
diameter CD in N; through Q and N draw QM, NP respectively
parallel to CD and AB and intersecting in P. Then the locus of
P is a cubic called the witch of Agnesi*.
Let A be the origin, QAM= 0; then
\(y2 + a2) cos2 0 = a2,
Qc and x = 2a cos2 8,
whence the equation of the curve is
/ p\ P(y2 + a2)x = 2as.........(1).
A L - -O0 M B The form of the curve is shown in
the figure. It has two real points of
inflexion at C and D and a third real
point at infinity; also the curve cuts the
D axis of x at right angles at B, and the
axis of y is an asymptote.
The curve has also a conjugate point
at infinity, which lies on the axis of x. This result at once follows
from (34) of ~ 48, from which we see that the nodal tangents are
determined by y2 + a2 = 0, and are therefore imaginary.
* Agnesi, Istituzione analitiche, Milano 1748. Loria, Bibliotheca math. 1897,
p. 7.
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