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

180

SPECIAL QUARTICS.

F2 as well as Q coincide with F; and the locus of P is the
elliptic limafon
r (c2- a2) = 2ac (c - a cos 0),
the two foci of which are F and F1. The point F is a conjugate
point and also a double focus.
When 1 > m > a/c, the value of c' becomes negative and F2
lies to the left of F, so that F now becomes the middle focus and
the oval (16) becomes imaginary. To find the conjugate oval, we
observe that Q now lies on PF produced, whence it can be shown
as before that the locus of Q is the conjugate oval
r- mr = -a........................(17)
which is one of the curves belonging to Case III.
When m = 1, F2 moves off to - c, whilst (15) becomes the
right-hand branch and (17) becomes the left-hand one of a hyperbola, whose foci are F and F1, and whose major axis is a.
When m > 1, c' and also c' -c are both positive, and therefore
F2 lies on the right of F, and is therefore the external focus.
The locus of P is given by (15), which may be written
r, + a/m = r/m.................... (18).
To find the conjugate oval, produce PF1 to R so that.
FR = a/rm; draw FQ so that the angle QFR = RFP. Then by
(18), FP/m = F1P + F1R = PR; whence
FP FQ
m - PR QR;
accordingly the locus of Q is
FQ - mFQ = - a,
which belongs to Case III.
Also if F1FR =, PFR = RFQ = a, it can be shown as before
that
F1P. F2Q= c2 (c a2)
and therefore Q is a point on the conjugate oval.
The oval of Descartes belongs to species VI, for which Pliicker's.
numbers are n= 4, 8=0, K= 2, m=6, r= 1, t=8; whilst the
two lima9ons belong to species IX, for which n = 4, 8 = 1, K = 2,
m=4, 7r=, ^=2.