414]
ON POLYZOMAL CURVES.
559
in the general case where the points A, B, G, D are not on a circle, this is, as has
been seen, a sextic curve, the locus of the foci of the conics which pass through
the four given points; in the case where the points are in a circle then the sextic
breaks up into two cubics (viz., observing that the curve under consideration is
VZ21 + Vra33 + Vw(£ + Vp3) = 0, where Vi : Vm. : Vw : Vp = a : b : c : d, these values do
of themselves satisfy the condition of decoinposability — + - + 3 = 0), that is, the
locus of the foci of the conics which pass through four points on a circle is composed
of two circular cubics, each of them having the four points for a set of concyclic
foci. It is easy to see why the sextic, thus defined as a locus of foci, must break
up into two cubics; in fact, as we have seen, the conics which pass through the four
concyclic points A, B, C, D have their axes in two fixed directions; there is con
sequently a locus of the foci situate on the axes which are in one of the fixed
directions, and a separate locus of the foci situate on the axes which lie in the other
of the fixed directions; viz., each of these loci is a circular cubic.
200. Adopting the notation of No. 188, or writing
RA = a 1 , RB = b 1} RC = c 1 , RD = d lt
(and therefore Z^Cx
= ttidi) we have
a : b
: c : d = — dx (61 — Cx)
: Cx
(a x — dx) : — 61 (aj — dj)
Moreover
VS = a + d
, VS = a + d,
Vm x = b + ,y/
^bed
a
, Vm 2 = b-^/-° d ,
V^ = C — >y/
bed
a
, - /bed
, vw 2 = c + y —-,
and we have
bed
a
(a } - dO 2 = af (cq - d1) 2 , a/— = -a 1 (a 1 - d x ) suppose;
di v a
and thence
that is
VS = (Oj - (h) (Jh - Cl), V4 = (Oi - dj) ( Ih-Ci)
^m 1 = (a 1 — dj) (Ci - Oj), Vm 2 = (a x — dj) ( c 2 + cq)
Vwj = (a x - d x ) (oj -61), Vw 2 = (oj - dO (- ch - 61),
VS : Vm a : l '/n 1 = b 1 — Cj : c x — a x : a 1 — b 1 ,
V/ 2 : Vra 2 : Vn 2 = 6 1 — c x : c x + (h : —a l — b l ,
agreeing with the formulae No. 188.
The tetrazomal curve
— dj (61 — Cj) VS] + Cj (aj - dO V^S — (oq — dj) Vg + oq (61 - Cx) ViD = 0
