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

THE OVAL OF DESCARTES.

175

sin a PR
But sn = IPP
sin a,
~~and ~sin 0 _a
and=
sin R c
a2 -- 2/?
whence P. FQ = 1
Accordingly the rectangle FP. FQ is constant; if therefore the
circle circumscribing PQF1 cut FF, in F2, and FF =c', it follows
that
FP.FQ =FF1.FF....................(5),,2 -- rn2C2
c'=-FF,= _ _ -.,) c(v-m)....(6),
and consequently PF is a fixed point.
262. To prove that F2 is the third focus of the curve.
Let F2P=r2; then since the triangles F2FP and QFF1 are
similar
r2 F1Q QR FQ-a
c' FQ =n-FQ mFQ
But r. FQ = cc', whence
r + r2mc/a = cc'/a..........().......(7).
Eliminating r between (3) and (7) we get
r2c/a - r, = (cc' - a2)/ma..................... ).
From (7) and (8) it follows that a relation similar to (2) exists
between FP and F2P, and also between F1P and F2P; whence F2
is a third focus of the inner oval P. In the same way it can be
shown that F2 is a third focus of the outer oval Q.
263. If PF1 be produced to meet the outer oval in Q', then
F1P. FQ' = F2F. F1F2.
Produce F1P to R' so that F1R' = a/m; then since
F1P + PR' = a/m,
and FP + mFP = a,
we obtain mPR' = FP.
Similarly mQ'R' = FQ',
FQ' FP
whence QP PR
Q'R' - P-' = m,