ON POLYZOMA.L CURVES. 
549 
414] 
and for either set of values the verification of the relation 
dap (/3 — 7) + d0a (y — a) -f dyr (a — 0) = 0, 
will depend on the two identical equations 
sin A sin {B — G) + sin B sin (G — A) + sin G sin {A — B) — 0, 
cos A sin (B — G) + cos B sin (G — A) + cos G sin (A — B) = 0 : 
although the foregoing solution for the case of a circular cubic is the most elegant 
one, I will presently return to the question and give the solution in a different form. 
Article No. 186. Focal Formulae, for the Symmetrical Curve. 
186. In the symmetrical case, where the foci A, B, G, D are on a line, then if, 
as usual, a, h, c, d denote the distances from a fixed point, we have the expressions 
of (a, b, c, d) in a form adapted to the formulae of No. 49, viz., 
a : b : c : d = (b — c)(c-d)(d-b): — (c — d)(d—a)(a—c): (d—a)(a—b)(b—d): — (a—b)(b—c)(c—a), 
so that, assuming 
the equation 
l : m : n = p(b —cf : a (c — a) 2 : t (a — b) 2 , 
l m n _ 
- + , + - = °, 
a b c 
becomes 
p (b — c) (a — d) + a (c — a) (b — d)+ t (a — b)(c —d) = 0, 
and the equation of the curve may be presented under any one of the four forms 
( . , ^r(d-c), Vi (b — d), Vp (c -b) )(v / 31, V©, \f(s, >/$) = <). 
vV (c — d), . , Vp (d — a), Vo- (a — c) 
l da(d — b), Vp (a — d), . , Vt(6— a) 
dp (b — c), Va(c — a), dr (a — b), 
Article No. 187. Case of the Symmetrical Circular Cubic. 
187. For a circular cubic we must have 
p (b — c) (a — d) + a (c —a) (b — d) + r (a — b) (c — cl) = 0, 
d p(b — c) + da(c — a) +dr(a — b) =0. 
These equations give dp‘da:dr = 1:1:1 (values which obviously satisfy the two 
equations), or else 
dp : dd : Jp = a + d-b-c : b + d — c — a : c + d — a — b.