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

THE OVAL OF DESCARTES.

177

cannot have four double points. Whence the oval (3) lies inside
the oval (4), and the foci F and F1 lie within both ovals.
C A F F/ B D F2
Let the oval (4) cut FF, in C and D, where C lies on the left
ofF: then
FD= a - me F F - mm)(- c)
-mr' c(1 - n2)
whence FF2 > FD and therefore F2 lies outside both ovals.
The three equations (5), (9) and (10) are fundamental ones in
the theory of these curves. The first one shows that each oval-is
the inverse of the other with respect to the internal focus F; the
second shows that either oval turned through two right angles is
the inverse of the other with regard to the central focus F1;
whilst the third shows that each oval is its own inverse with
respect to the external focus F2. Also the two tangents drawn
from F2 to the inner and outer ovals respectively are equal.
A pair of ovals which possess these focal properties are called
conjugate ovals; and their forms when a > c, m < 1 are shown in
the figure.
266. When m = 1, F1P = PR, see figure to ~ 26i, whence the
angle
PRF= PFiR = RF1Q;
accordingly F1Q is parallel to FP, also from (6) FF2 = o. In this
case every point of the outer oval and also the external focus
B. C. 12