667] 
ON THE BICIRCULAR QUARTIC. 
225 
fact, functions of the five quantities f+ 0, f—g, 0 x — 0, 0» — 0, 0 3 —0)\ and we can 
in terms of these data express the equations as well of the dirigent conics as of 
the circles of inversion; viz. taking X, Y as current coordinates, the equations are 
j^-0+X-g =1, (X — a ) 2 + (F — /3 ) 2 - 7 s =0, or X 2 + F 2 -2aX-2/3 Y + k =0, 
/Tk + ¿Te, = >■ (X-*,y + <Y-ft) 2 - 7> 2 = 0, or X* + F 2 - 2a,X - 2/3, F + k, = 0, 
JTe, + jvs., = 1 -( X ~ + (' Y - - * a = 0, or X 2 + F 2 - 2<r„X - 2ft r+ k, « 0, 
7+0 + g~+6 = 1 ' * X - **)* + (F — /3,,)= -= 0, or X 2 + F 2 - 2<t,X - 2/3, F + k, = 0, 
where 
\J f± 6 -l±^I±ild±i! = (/+ 0) « = (/+ 0.) a, = (/+ 0.) a, = (/+ 0,) 
^g + e.g + e,.g + e,.g+e, = (g + 0)/3=(g+ gj p i = {g+ ft) ft=(J, + 0,) ft, 
y+ 0 . g + 0 . 7 2 = 0 — 0 x . 0 — 0. 2 .0 — 0 3 , 
f+ 0 x .g + 01. Yj 2 = 0, - 0 • 0] - 02-01 - 03, 
/ + 0o . g + . Y2 2 =02—0 • #2 - #1 ■ #2 — 
/+ • g + 0 3 • Ys 2 = 0.-0 .03- 01-03 - 02, 
f+ g + 0 + 01 + 01 + 03 = Jc + 20 = k x + 20! = Jc 2 + 202 = k 3 + 20 3 . 
3. The geometrical relations between the dirigent conics and circles of inversion 
are all deducible from the foregoing formulae; in particular, the conics are confocal, 
and as such intersect each two of them at right angles; the circles intersect each 
two of them at right angles. Considering a dirigent conic and the corresponding 
circle of inversion, the centres of the remaining three circles are conjugate points in 
regard as well to the first-mentioned conic, as to the first-mentioned circle; or, 
what is the same thing, they are the centres of the quadrangle formed by the 
intersections of the conic and circle. 
4. The centre of the conics and the centres of the four circles lie on a 
rectangular hyperbola, having its asymptotes parallel to the axes of the conics. Given 
the centres of three of the circles (this determines the centre of the fourth circle) 
and also the centre of the conic, these four points determine a rectangular hyperbola 
(which passes also through the centre of the fourth circle); and the axes of the 
conics are then the lines through the centre, parallel to the asymptotes of the 
hyperbola. 
C. X. 
29