414] 
ON POLYZOMAL CURVES. 
545 
equation involving the parameters of the given circles, and also d, di', the parameters of 
the variable circle; viz., an equation determining di', the radius of the variable circle, 
in terms of d, the coordinate of its centre. I consider in particular the case where 
the given circles are points ; that is, where the given equation is 
V ¿21 + V to23 + V = 0. 
The equations here are 
a -j- b +0 = — d, 
aa + b b + cc = — d d, 
a a 2 + b b- + cc 2 = — d (d 2 — d" 2 ), 
and from these we obtain 
a (a — 6) (a — c) = — d ((d — b) (d — c) — d" 2 ) 
b (6 — c) (6 — a) = — d ((d — c) (d — a) — d" 2 ) 
c (c — a) (c — b) = — d ((d — a) (d — b) — d" 2 ), 
so that the equation - + ~ + - = 0 becomes 
cl D C 
l(a — b) (a— c) to(b — c)(b — a) n (c — a)(c — b) _ 
(d-b) (d-c)- d" 2 + (d-c) (d - a)^d" 2 + (d-a) (d-b)-d" 2 ~ ’ 
or, as this is more conveniently written, 
l 1 to 1 n l_n 
b — c (d — b)(d — cj — d" 2 c — a (d — c) (d — a) — dl' 2 a — b (d — a)(d — b) — d" 2 ’ 
viz., considering d, di' as the abscissa and ordinate of a point on a curve, and repre 
senting them by x, y respectively, the equation of this curve is 
l 1 to 1 n 1 _ 
b — c (x — b) (x — c) — y 2 c — a (x — c) (x — a) —y 2 c — a (x — a) (x — b) —y 2 ’ 
which is a certain quartic curve ; and we have the original curve 
V/21 + Vm23 4- V?i(£ = 0, 
as the envelope of a variable circle having for its diameter the double ordinate of 
this quartic curve. 
Write for shortness —, ~~ l —, M, N respectively, then the equation 
0 — C C CL CL 0 
of the quartic curve may be written 
1L [(# — a) 2 (x — b)(x — c) — y 2 (x — a) (2x — b — c) + y 4 ] = 0, 
viz., this is 
XL [x (x — a) (x — b) (x — c) 
— y 2 (2x 2 — (a + b + c)x + (ab + ac + 6c)) + y 4 
— a (x — a) (x — b) (x — c) + y 2 (ax + be)] = 0, 
C. VI. 
69