540 
^ ON POLYZOMAL CURVES. 
[414 
then the equation of the envelope is 
that is, it is 
(l 2 , m 2 , n 2 , — mn, — nl, — lm\ 51°, 53°, (S 0 )- = 0 ; 
(1, 1,1,- 1, - 1, - 1 #m°, m23°, w6°) 2 = 0, 
or, what is the same thing, it is 
Vm° +Vm33° +VwÌ° = 0. 
172. It has been seen that the equations of the nodal tangents at the points 
1, J respectively are respectively 
Vi (£ — clz ) + Vm (£ — /3z ) + V?i (£ — yz ) = 0, 
Vi (rj — a'z) + Vm (y — ¡3'z) + Vw (rj — y'z) = 0, 
and that these are the equations of the tangents to the conic Ivw + muni + nuv = 0 
from the points I, J respectively. We have thus Casey’s theorem for the generation 
of the bicircular quartic as follows:—The envelope of a variable circle which cuts at 
right angles the orthotomic circle of three given circles 21° = 0, 33° = 0, QC = 0, and has 
its centre on the conic Iviu + mwu + vuv = 0 which passes through the centres of the 
three given circles is the bicircular quartic, or trizomal 
Vm° + Vm33° + Vw(S° = 0, 
which has its nodo-foci coincident with the foci of the conic. 
173. To complete the analytical theory, it is proper to express the equation of 
the orthotomic circle by means of the areal coordinates (u, v, w). Writing for shortness 
a 2 + a 2 — a” 2 = a, &c., and therefore 
then if as before 
21° = x 2 + y 2 — 2axz — 2a'yz — (Cz 2 , &c., 
u : v : w — 
x, y, z 
x, y, z 
x, y, z 
b, V, 1 
c, c, 1 
a, a, 1 
c, d, 1 
a, a, 1 
b, V, 1 
and therefore 
x : y : z = au + bv + cw : a'u + b'v + c'w : u + v + w, 
the equation of the orthotomic circle is 
x — az, y — a'z, ax + a'y — ciz = 0, 
x — bz, y — b'z, bx + b'y — b'z 
x — cz, y — c'z, cx + c'y — dz 
viz., throwing out the factor z, this is 
u(ax + a'y — az) + v (bx + b'y — b'z) + w (cx + c'y — c'z) = 0,