536 
ON POLYZOMAL CURVES. 
[414 
or substituting for fi, v, their values, it is 
l (a - /3) (a - 7 ) + m (/3 - 7) (/3 - a) + n (7 - a) (7 - ¡3) = 0, 
or, as it is more simply written, 
l m n 
+ —— + « = 0. 
/3 — y 7 — a a — /3 
163. If the node at (77 = 0, 0= 0) be also a cusp, then we have in like manner 
l m n _ (\ 
¡3' — <y'+ y — a.' + a' — ¡3' 
Now observing that 
(7-a) (#'-£')-W-«')(«-/3), = a, a', 1 
/3, /3', 1 
7. 7. 1 
= (a - /3) (/3' — 7 0 — (a' — ¡3') (/3 — 7), 
= (/3 - 7) (7 - «') - (£' - 7) (7 - a )> 
= il suppose: the two equations give 
l : m : n = il (/3 — 7) (/3' — 7') : O (7 — a) (7' — a') : il (a — /3) (a' — /3'); 
or if il is not = 0, then 
l : m : n= (/3 - 7) (/3' - 7) : (7 - a)(7 - a') : (a - /3) (a' -/3'). 
164. If 
n = 
or, what is the same thing, if 
a, a', 1 
/3, /3', 1 
7> 7; 1 
a, a', 1 
5, 6', 1 
c, c', 1 
, =0, 
= 0, 
the centres A, B, G are in a line ; taking it as the axis of x, we have a = a! — a, 
¡3 = (3’ = b, 7 = 7' = c ; and the conditions for the cusps at 7, J respectively reduce 
themselves to the single condition 
l m n 
t 1 H i = 0, 
b—c c—a a—b 
so that this condition being satisfied, the curve 
Vi {{x - azf + if - a'V} +Vm{(x- bzf+ if - 6'V} + \/n{(x — czf + y-- c" 2 z 2 } = 0 
is a Cartesian; viz., given any three circles with their centres on a line, there are 
singly infinite series of Cartesians, each touched by the three circles respectively; 
a