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

148

BICIRCULAR QUARTICS.

In the figure to ~ 200, let (f', g') be the coordinates of the
centre A of the circle U referred to 0, and (fl, g1) of A referred
to E; also let (:, v) be current coordinates referred to A, the axes
being parallel to those of the focal conic. Then
f=f+f', g =g+g,...............
Also let
P =ff'+ gg' + "27
SI - ~2 + q2 + 2fi + 2gq + r2........
where A02 = r12 + 82. Then (12) can be transformed into
(S1 + 2P)2 = 4 {2 ( ( +/')2 + b2 (r + g')2},
or 12 = 4 {[2 (a2 - P) + 72 (b2 - P)
+ 2 (a2f'- Pf) + 2r (b2' - Pg)
+ a2f/2 + b2g'2 - P12 - P2}...................(28).
The equation
(+)2 ( + ( 1 (+ +{( f') + (a + g')2 82} = 0
represents a conic passing through the points P, Q, R, S in which
the focal conic intersects the circle of inversion; and by suitably
determining X this conic may be made to represent the two
straight lines AP, AQ. Since A is the origin, we must determine X from the conditions that the coefficients of t and r and
also the absolute term vanish, which give
+ '= 0........................(29),
a2.(2)
|b + xg'=........................ (30)
+ - + x (f + / 2 _ 2............(31)
Multiplying (29) and (30) byf,, g,, adding, and taking account
of (:31) and the first of (27) we obtain
P + 1 = 0...........................(32),
whence (29) and (30) become
Pf -a2f'=, Pg- b2g'=0............(33),
which show that the coefficients of | and x in (28) vanish.