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

THE OVAL OF DESCARTES. 179
produce F1P to R so that F1- = a/m, and on F1R take a point Q
such that angle QFR = RFP; then since by (3) the locus of P
may be written F1P + FP/ln = a/m, it follows that RP = FP/m,
whence
RP RQ 1
FP FQ m'
and
F1Q - FQ/nm = FJR = a/m,
and therefore the locus of Q is the oval
r - mr = - a...................... (14).
It can also be shown as in ~ 261 that
a2 2
FIP. FQ-m- 1 = F. FF,
and consequently Q is a point on the conjugate oval, which by
(14) belongs to Case IV.
Cases II. and III.
267. In Case II, c > a > O, a/c > m > -; but we shall find
it convenient to begin by discussing the two ovals
r - mr = a...........................(15),
r + mrn = a...........................(16),
in which m is a positive quantity lying between a/c and 0.
In the figure FR = a, whilst P and Q are two points such that
RP RQ p
= ~-= ~rn, R
F1P F1Q,
whence (15) and (16) are the equations of
the loci of P and Q respectively. We can
also prove as before that F -2 F,
FQ. FP = FF. FF= cc' a - n
accordingly as long as m lies between 0 and a/c, the value of F2 is
positive and less than c, whence F2 is the central focus and F the
external one.
When m = 0, both ovals coalesce into a circle of radius a
whose inverse points are F2, Fl. When mc c=a, FF2=0 and

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