414] 
ON POLYZOMAL CURVES. 
531 
151. Consider the case of the bicircular quartic, and take as before (£ = 0, 2 = 0), 
and (77 = 0, z = 0) for the coordinates of the points I, J respectively. Let the two 
tangents from the focus A be f — az = 0, 77 — a'z = 0, say for shortness p = 0, p' = 0, 
then the equation of the curve is expressible in the form pp'U=V 2 ( 1 ), where TJ= 0, 
V = 0 are each of them a circle, viz., U and V are each of them a quadric function 
containing the terms z 2 , zi7, zg, and £77. Taking an indeterminate coefficient A, the 
equation may be written 
pp' (U + 2\V+ X 2 pp') = (V + Xpp') 2 , 
and then A may be so determined that U + 2XV+X 2 pp'= 0, shall be a 0-circle, or 
pair of lines through I and J. It is easy to see that we have thus for X a cubic 
equation, that is, there are three values of A,, for each of which the function 
U + 2AF 4- X s pp' assumes the form (£ — /3z) (77 — fi'z), = qq' suppose: taking any one of 
these, and changing the value of V so as that we may have V in place of V+ Xpp', 
the equation is pp'qq' + V 2 , where V = 0 is as before a circle, the equation shows that 
the points of contact of the tangents p = 0, p' = 0, q = 0, q' = 0 lie in this circle V=0. 
The circumstance that X is determined by a cubic equation would suggest that the 
focus q = 0, q' = 0 is one of the three foci B, C, D concyclic with A ; but this is 
the very thing which we wish to prove, and the investigation, though somewhat long, 
is an interesting one. 
152. Starting from the form ppqq' = V 2 , then introducing as before an arbitrary 
coefficient X, the equation may be written 
pp (qq + 2XV + X 2 pp') = ( V+ Xpp') 2 , 
and we may determine A, so that qq'+ 2XV + X 2 pp'= 0 shall be a pair of lines. 
Writing V = Hgr) — Lrjz — Lljz + Mz 2 , and substituting for pp and qq their values 
(f — oz)(t7 — a'z) and (£ — ¡Hz) (77 — fi'z), the equation in question is 
(1 + 2XH + A, 2 ) £77 - (/3 + 2XL + X 2 *) 77z - (/3' + 2XL' + AV) %z + (/3/3' + 2Ailf + AW) * 2 = 0, 
and the required condition is 
(1 + 2XH+X 2 ) (/3/3' + 2XM + A 2 W) = (/3 + 2A L + A 2 «) (/3' + 2A L' + A 2 «); 
or reducing, this is 
(:2M + 2H/3/3’ - 2L'/3 - 2Z/3) 
+ A ((a — /3) (a' — /3') + \HM— 4LL') 
+ A 2 (2M + 2Hact! - 2L'a - 2La!) = 0, 
viz., A is determined by a quadric equation. Calling its roots A 1( and A 2 , the foregoing 
equation, substituting therein successively these values, becomes (£ — 72) (77 — y'z) = 0, and 
(£ — 8z) (rj — 8'z) = 0 respectively, say rr = 0 and ss' = 0. 
1 This investigation is similar to that in Salmon’s Higher Plane Curves, p. 196, in regard to the double 
tangents of a quartic curve. 
67—2