414] ON POLYZOMAL CURVES. 547 
Article Nos. 182 and 183. Focal Formula} for the General Curve. 
182. Considering any three circles centres A, B, C, and taking 21°, &c., to denote 
as usual, let the equation of the curve be 
vm° + Vt^T + \№ = 0 ; 
then considering a fourth circle, centre D, a position of the variable circle, and having 
therefore the same orthotomic circle with the given circles, so that as before 
a2(° + b‘0° + cr + d3)° = 0, 
the formulae No. 47 (changing only U, V, W, T into 21°, 33°, (S°, 2)°) are at once 
applicable to express the equation of the curve in terms of any three of the four circles 
A, B, C, R 
In particular, the circles may reduce themselves to the four points A, B, C, D, a 
set of concyclic foci, and here, the equation being originally given in the form 
V ¿21 + V m23 + V w(5 = 0, 
the same formulae are applicable to express the equation in terms of any three of 
the four foci. 
183. It is to be observed that in this case if the positions of the four foci are 
given by means of the circular coordinates (a, -, lY &c., which refer to the centre of 
the circle ABCD as origin, and with the radius of this circle taken as unity, then 
the values of a, b, c, d (ante, No. 90), are given in the form adapted to the formulae 
of No. 49, viz., we have 
a : b : c : d = a. (ßy8) : — ß (y8a) : y (8aß) : — 8 (aß7), 
where (ßy8) = (ß — 7) (7 — 8) (8 — ß), &c. The relation ~ + ^ + - = 0, putting therein 
a d c 
l : m : n — pa (ß — 7) 2 : aß(y — a) 2 : ry (a — ß) 2 , (or, what is the same thing, taking the 
equation of the curve to be given in the form (ß — y)*JpaA + (7 — a) Vo-/323 +(a — ß)fry(^ = 0), 
p (ß ~ 7) (a - S) + a- (7 - a) (ß - 8) + r (a - ß) (7 - 8) = 0, 
viz., this equation, considering p, a, t, a, ß, 7 as given, determines the position of the 
fourth focus D, or when A, B, C, D are given, it is the relation which must exist 
between p, a, t ; and the four forms of the equation are 
( . , fr (8 -7), fa(ß-8), \Tp (7-/3) ) (Va2T, V/323, VS3)) = 0, 
Vt (7 — 8), . , Vp(8— a), Va (a— 7) 
\'o-(8—ß), fp(a—8), . , fr(ß-a) 
Vp (/3—y), Vo-(7-a), fr(a-ß), 
viz., the curve is represented by means of any one of these four equations involving 
each of them three out of the four given foci A, B, C, D. 
69—2