546
ON POLYZOMAL CURVES.
[414
or what is the same thing, the equation is
(L + M + N)[x(x — a) (x — b)(x — c) — y 2 {2x 2 — (a + b+ c) x + ab + ac + bc) + y 4 ]
— (La + Mb + Nc) (x — a) (x — b)(x — c)
+ if {(La + Mb + Nc) x + Lbc + Mca + Nab] = 0.
In the particular case where L + M + N = 0, that is, where
l m . n .
t 1 1 r — 0,
b — c c — a a — b
the quartic curve becomes a cubic, viz., putting for shortness
£ _ Lbc + Mca + Nab
~ La + Mb + Nc ’
the equation of the cubic is
„ _ (x - a) (x — b) (x — c)
y- _ ,
viz., this is a cubic curve having three real asymptotes, and a diameter at right
angles to one of the asymptotes, and at the inclinations + 45°, — 45° to the other
two asymptotes respectively—say that it is a “ rectangular ” cubic. The relation
= 1———|— 7 ^— 1 =0 implies that the curve V¿21 + Vm s $ + V?i(S = 0 is a Cartesian, and
b — c c — a a — b
we have thus the theorem that the envelope of a variable circle having for diameter
the double ordinate of a rectangular cubic is a Cartesian.
I remark that using a particular origin, and writing the equation of the rectangular
2A
cubic in the form f — a? — 2mx + a. + - , the equation of the variable circle is
2A
(x — d) 2 + f = d 2 — 2md + a + ,
that is
2A
x 2 + f — a — 2d (x — m) = 0,
where d is the variable parameter. Forming the derived equation in regard to d, we
have
A
and thence
„ 9 4 A
or + y 2 - a = -J-
(x? + y 2 — a) 2 = = 16.4 (x — m),
that is, the equation of the envelope is (x 2 + y 2 — a) 2 = 164 (x — m) = 0, which is a
known form of the equation of a Cartesian.