560 
ON POLYZOMAL CURVES. 
[414 
is thus decomposed into the two trizomals 
(Zq — Ci) V$l + (Cj — cq) V53 + (cq — Zq) Vg = 0, 
(6, - cd VI + (C! + ad V$ ~ («1 + Zq) VI = 0. 
201. Observe that the tetrazomal equation is a consequence of either of the 
trizomal equationstaking for instance the first trizomal equation, this gives the 
tetrazomal equation, and consequently any combination of the trizomal equation and 
the tetrazomal equation is satisfied if only the trizomal equation is satisfied. Multiply 
the trizomal equation by — cq + d x and add it to the tetrazomal equation; the resulting 
equation contains the factor cq, and omitting this, it is 
(6i - Ci) (- VI + VS)) + (cq - dd (V© - VI) = 0, 
where observe that Zq — Ci is the distance BC, and cq — d x the distance AD. But in 
like manner multiplying the second trizomal equation by — a 1 + d l , and adding it to 
the original tetrazomal equation, the resulting equation, omitting the factor cq, is 
(Zq - Ci) (- VI + V®) - («i - dd (VS - VI) = 0; 
viz., it is in fact the same tetrazomal equation as was obtained by means of the first 
trizomal equation. 
The new tetrazomal equation, say 
(Zq - Cl) (- VI + VS)) + («1 - dd (Vs - VI) = o, 
is thus equivalent to the original tetrazomal equation; observe that it is an equation 
of the form VZl + VraS + V?il + VpS) = 0, where 
VZ = —(6i —Ci), Vm = cq — cZj, V?i = («i — dd, Vp = Zq-cq, 
and where consequently VZ + Vp = 0, Vm + Vw=0, that is an equation of the form 
(198) III., decomposable, as it should be, into the equations of two circular cubics. 
Writing 
-VI+VS) . VS-VI 
Ui -d x ~ ’ Zq — Ci “ ^ 
where ^ is an arbitrary parameter, the curve is obtained as the locus of the inter 
sections of two similar conics having respectively the foci (A, D) and the foci (B, C) 
(see Salmon, Higher Plane Curves, p. 174): whence we have the theorem, that if 
A, B, G, D are any four points on a circle, the two circular cubics which are the 
locus of the foci of the conics which pass through the four points A, B, C, D, are 
also the locus of the intersections of the similar conics, which have for their foci 
{A, D) and (B, G) respectively; and of the similar conics with the foci (B, D) and 
(G, A) respectively; and of the similar conics with the foci (G, D) and (A, B) respectively. 
202. Y. VZ = Vm = Vw = Vp. The order of the tetrazomal is = 5, whence those 
of the trizomals should be = 3 and = 2 respectively. To verify this observe that the