548 
ON POLYZOMAL CURVES. 
[414 
Article Nos. 184 and 185. Case of the Circular Cubic. 
184. In the case of a circular cubic, we must have 
P “ 7) ( a ~ $) + o- (7 - a) (¡3 - 8) + t (a - /9) (7 - 8) = 0, 
Vap (¡3 — 7) + V/3cr (7 — a) 4- V77- (a — /3) = 0, 
which, when the foci A, B, C, D are given, determine the values of p : cr : t in order 
that the curve may be a circular cubic. We see at once that there are two sets of 
values, and consequently two circular cubics having each of them the given points 
A, B, C, D for a set of concyclic foci. The two systems may be written 
Vp : Ver : Vt = VaS — V/S7 : f ¡38 —fya. : V78 —Va/3, 
viz., it being understood that Va.8 means Vac.VS, &c., then, according as f8 has one 
or other of its two opposite values, we have one or other of the two systems of 
values of p : a : r. To verify this, observe that writing the equation under the form 
f ap : V/3a : fyr — af 8 — Va/3y : ¡3 f8 — f a.f3y : 7 ^8 — V a/3y, 
the second equation is verified; and that writing them under the form 
p : a : r = — (¡3 4- 7) (a 4- 8) + M : — (7 4- a) (¡3 4- S) + M : — (a 4- /3) (7 4- 8) + M, 
where 
M = /87 + a.8 4- yet. 4- /38 + a/3 + y8 — 2 V aj3y8, 
the second equation is also verified. 
185. If we assume for a moment a = cos a + i sin a = e ia , &c., viz., if a, b, c, d be 
the inclinations to any fixed line of the radii through A, B, C, D respectively, then we 
have 
f (X.8 4 VBy — (a+b+c+dji jgi(a+ci—b—c)i _j_ q—\(a+d—b—c)i| 
Va (/3—7) = gh(a+b+c)i c)i g—i(b—c)i | ( 
and thence 
\'<xp (/3 — 7) : V/3cr (7 — a.) : Vyr (a — (3) = cos £ (a + d - b — c) sin |(6 — c) 
: cos | (b 4- d — c — a) sin | (c — a) 
: cos \(c 4- d — a — b) sin \ (a — b); 
or else 
= sin J (a 4- d — b — c) sin \{b — c ) 
: sin £ (b 4- d — c — a) sin 4 (c — a) 
: sin l (c 4- d — a — b) sin \ (a — b). 
Putting in these formulae, 
i (a — b — c) = A, then we have B — C = ^ (b — c), 
l(b — c — a) = B, „ C — A = £ (c — a), 
\(c — a —b) = C, „ A -B = ± (a -b),