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

170

SPECIAL QUARTICS.

Let ABC be the triangle, N its orthocentre, 0 the centre of
the hyperbola. Then N is a fixed point and ON a fixed line;
L
B C
also since 0 lies on the nine-point circle of the triangle, and N is
the centre of the latter,
ON = NE = 'R.
Let OL be the asymptote, and S the focus of the hyperbola;
let NOS = >. Then if OE meet the hyperbola in P, OE= EL;
and from the equation of a rectangular hyperbola referred to its
asymptotes
0S2= 40P2 sin LOP cos LOP = 2OP2 sin OEC.
Also from the equation referred to a pair of conjugate
diameters
AE2 = OE2 - OP2;
but
AE = R /3, OE = 20N cos NEO = R sin OEC,
whence
OS2 = 1R2 (4 sin OEC' - 3 sin OEG)
=- R' sin 0OEC.
But OEC= 2LOE = 2 (- r - OEC),
whence OS2 = R2 sin 2),
and therefore the locus of S is a lemniscate.
256. To find the equation of the evolute of a lemniscate.
In the figure to ~ 253, let 0 be the centre of curvature at
P, Q the corresponding point on the rectangular hyperbola which
is the first negative pedal of the lerniscate. Let the normal at
P meet CQ in K. Then
= QCA = A CP = CPK,
whence KC = KP = KQ.