An elementary treatise on cubic and quartic curves, by A. B. Basset.

146

BICIRCULAR QUARTICS.

and (23) becomes
U=S- Ia cosA &c.
The constant k is determined from the fact that the point A is
the centre of U, and therefore the pole of the line at infinity;
whence k= 2 and (23) becomes
U = S- 2a cos A)
V = S - 21 cos B................... (24).
W = S- 2Iy cos C)
215. If X, tu, v be variable parameters, the circle
xU+P V+ vW=O
cuts the circle S orthogonally.
The radical axes of any three circles intersect at the radical
centre; and from this point as centre a circle A can be described
cutting each of the three circles orthogonally. Also if a fourth
circle be described, such that the radical axis of the latter and
any one of the three circles passes through the radical centre,
this circle will be cut orthogonally by S. Hence the circle
XU+/LV+ vW=v will be cut orthogonally by S, provided the
radical axis of itself and U passes through 0. Now
XU+ /V+ - v W = (X + + v) S - 21 (Xa cos A + /u/ cos B + vy cos C),
and the radical axis of this and U is
- (,/ + v) a cos A +,/3 cos B + v7 cos C =0,
which obviously passes through the point 0, where
a cos A =, cos B = y cos C.
From (24) it follows that the equation of a bicircular quartic
may be expressed in the form
I (S - 2a cos A)2 + mn (S - 219 cos B)2
+n(S- 27 cos C)2 =0......(25),
which shows that the quartic passes through the two circular
points at infinity.
The condition that the focal conic (22) should be a parabola is
that 1 + m + n = 0, in which case (25) becomes
S (la cos A + m,3 cos B + ny cos C)
= I (la2 cos2 A + n/32 cos2 B + n72 cos2 C),
which is the equation of a circular cubic.