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.