538 
ON POLYZOMAL CURVES. 
[414 
or in the circular coordinates (g, ij, 2=1), if (a, a', 1), (/3, ft, 1), (7, 7', 1) be the 
coordinates of the three points respectively, then 
u : v : w = 
£> v » * 
£» v, z 
£, V , z 
ft ft, 1 
y, ft, 1 
a, cl, 1 
7> ft, 1 
a, a', 1 
ft ft, 1 
167. For the point I we have (g, 77, z) = (0, 1, 0), and hence if its areal 
coordinates be (u 0 , v 0 , w 0 ), we have 
u 0 : v 0 : w 0 = ¡3 — y : 7-a : a —ft 
and hence also, (u, v, w) referring to the current point P, we find 
v 0 w — WqV = (7 — a) [(a' - ft) (g - clz) - (a - ft) (77 - a'z)] 
if 
whence 
-(«-£) [(7 - *') (g - az) - (y - a) ( v - clz)], = il (g — clz), 
ft = (7 - «) (« “ ft) -(«-£) (7' - «'), 
a, a', 1 
ft ft. 1 
7’ 7» 1 
v 0 w -w 0 v : w 0 u — wu 0 : îî 0 v — = g — a z : g — /3 z : £ — 7 2, 
and in precisely the same manner, if u 0 ', v 0 ', w 0 ' refer to the point J, then 
and 
V 0 'iv — W 0 'v 
u 0 ' : Vo : Wq = ft — ft : ft — cl : cl — ft, 
: Wq'u — wuq : iftv - uvô = 77 — clz : 77 — /3'^ : 77 — ftz. 
168. Consider the conic 
(a, 6, c, f g, K$u, v, w) 2 = 0, 
where u, v, w are any trilinear coordinates whatever; and take the inverse coefficients 
to be {A, B, G, F, G, H) (A = bc —f 2 , &c.), then for any given point the coordinates of 
which are (n 0 , v 0 , w n ), the equation of the tangents from this point to the conic is, 
as is well known, 
{A, B, C, F, G, H^VoW — WoV, w 0 u — u 0 w, u () v — v 0 u) 2 = 0 ; 
consequently for the conic 
(a, b, c, f, g, K$u, v, w) 2 = 0, 
where (u, v, w) are areal coordinates referring, as above, to any three given points 
A, B, G, the equation of the pair of tangents from the point I to the conic is 
(A, B, G, F, G, H\g - clz, g - @z, g - ryz) 2 = 0, 
and that of the pair of tangents from J is 
(A, B, G, F, G, H\t) — clz, 77 — ftz, 77 —7^) 2 = 0, 
these two line-pairs intersecting, of course, in the foci of the conic.