414] 
ON POLYZOMAL CURVES. 
539 
169. In particular, if the conic is a conic passing through the points A, B, G, 
then taking its equation to be 
Ivw + mwu + nuv = 0, 
the inverse coefficients are as (lm 2 , ri\ — 2mn, — 2nl, — 2Im), and we have for the 
equations of the two line-pairs 
Vi (| — az) + Vm (£ — ¡3z ) + fn (g — yz ) = 0, 
\/l (77 — az) + Vm (77 — f3'z) + (77 — y'z) = 0. 
Article No. 170. The Theorem of the Variable Zomal. 
170. Consider the four circles 
21° = 0, 33° = 0, 6° = 0, 2)° = 0 (21° = (x-azf + {y- a!zf - a"*z\ &c.), 
which have a common orthotomic circle; so that as before 
where 
a 21° + b23° + eg 0 + d3)° = 0, 
a : b : c : d = BCD : - GDA : DAB : - ABC. 
I consider the first three circles as given, and the fourth circle as a variable circle 
cutting at right angles the orthotomic circle of the three given circles; this being 
so, attending only to the ratios a : b : c, we may write 
a : b : c = DBG : DGA : DAB, 
that is, (a, b, c) are proportional to the areal coordinates of the centre of the variable 
circle in regard to the triangle ABG. 
171. Suppose that the centre of the variable circle is situate on a given conic, 
then expressing the equation of this conic in areal coordinates in regard to the 
triangle ABG, we have between (a, b, c) the equation obtained by substituting these 
values for the coordinates in the equation of the conic; that is, the equation of the 
variable circle is 
a2P + b23° + eg 0 =0, 
where (a, b, c) are connected by an equation 
(a, b, c, f, g, h\a, b, c) 2 = 0. 
Hence (A, B, G, F, G, H) being the inverse coefficients, the equation of the envelope 
of the variable circle is 
(A, B, G, F, G, H\A°, 23°, g°) 2 =0, 
and, in particular, if the conic be a conic passing through the points A, B, G, and 
such that its equation in the areal coordinates (u, v, w) in regard to the triangle 
ABG is 
Ivw + mwu + nuv = 0, 
68—2