534 
ON POLYZOMAL CURVES. 
[414 
viz., we have the theorem that for a bicircular quartic if (£ — az = 0, rj — a'z = 0), 
— = 0, 7) — /3'z = 0, (£ — 72 = 0), Tj — y'z = 0) be any three concyclic foci, then the 
equation is as just mentioned; that is, the curve is a trizomal curve, the zomals 
being the three given foci regarded as 0-circles. The same theorem holds in regard 
to the circular cubic, and a similar demonstration would apply to this case. 
158. It may be noticed that we might, without proving as above that the two 
foci (p = 0, p' = 0), (q =0, q = 0) were concyclic, have passed at once from the form 
ppqq = V 2 , to the form X *Jpp + */qq' + K Vrr =0 (or V¿21 = Vm23 = Vw(I = 0), and then 
by the application of the theorem of the variable zomal (thereby establishing the 
existence of a fourth focus concyclic with the three) have shown that the original 
two foci were concyclic. But it seemed the more orderly course to effect the demon 
stration without the aid furnished by the reduction of the equation to the trizomal 
form. 
Part IY. (Nos. 159 to 206). On Trizomal and Tetrazomal Curves where the Zomals 
are Circles. 
Article Nos. 159 to 165. The Trizomal Curve—The Tangents at I, J, &c. 
159. I consider the trizomal 
V7r + Vm23° + VidT = 0, 
where A, B, C being the centres of three given circles, 21°, &c. denote as before, viz., 
in rectangular and in circular coordinates respectively, we have 
21° = (x - az) 2 + (y - a'z) 2 - a" 2 z 2 , = (£ - az) ( v - a'z ) - a" 2 z 2 , 
S3° = {x- hz) 2 +(y- h'z) 2 - b" 2 z 2 , = (f - /3z) ( V - /3'z) - b" 2 z 2 , 
®° = (« - cz) 2 + (y- cz) 2 - c" 2 z 2 , = (ij-yz) ( v - ry'z) - c'V. 
By what precedes, the curve is of the order = 4, touching each of the given circles 
twice, and having a double point, or node, at each of the points I, J\ that is, it is 
a bicircular quartic: but if for any determinate values of the radicals Vl, Vm, Vw, 
we have 
Vi +Vm -(- Vw =0, 
then there is a branch 
V W° + Vm23° + Vn(T = 0, 
containing (z= 0) the line infinity; and the order is here =3: viz., the curve here 
passes through each of the points I, J and through another point at infinity (that is, 
there is an asymptote), and is thus a circular cubic. 
160. I commence by investigating the equations of the nodal tangents at the 
points I, J respectively; using for this purpose the circular coordinates (£, 77, s = l), 
it is to be observed that, in the rationalised equation, for finding the tangents at 
(£ = 0, z = 0) we have only to attend to the terms of the second order in (£, z), and