PROFESSOR CAYLEY ON THE BICIRCULAR QUARTIC. 
443 
where 
^/T+FZIYStZ+5 = (/ + ^„=(/ + « 1 ) a -(/+fl 2 ) aa =(/+y« 3 , 
v /Hi5±£^E5±5=( S r+^e= (f+w,= (?+«g,=(?+fl.)p,. 
/+0 . <7+$ . y 2 = 3 6 -6 a . Q -0 3 , 
Z+ty+t / = a,-* .fl.-fl.-fl.-fl,, 
/+^2 • <7 + ^2 • 72 = ^2 _ ^ . 5 2 — 0! . 0 2 — 0 3 , 
/+^3 • #+03 • 7s = $3 — ^ . Q 3 — 0, . 5g — fi 2 . 
.Z+^+^+^i +^2+^3 == ^+25=^ 1 +2S 1 =& 2 -t-202=&3+2S 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. 
5. The equation of the bicircular quartic may be expressed in the four forms 
(X 2 +Y 2 -£ ) 2 -4[(/+(3 )(X— a y+(9+0 )(Y-3 ) 2 ]=0, 
(X 2 +Y 2 — kj- 4[(/+fl 1 )(X— ai )H(m)(Y-fr) 2 ]=0, 
(X 2 + Y 2 — & 2 ) 2 — 4 [(/+ 0 2 ) (X — a 2 ) 2 + ( <7 + 0 2 )( Y — /3 2 ) 2 ] == 0 , 
(X 2 + Y 2 — Jc 3 f — 4[(/+ 0 3 )(X — a 3 ) 2 + (<7 + 0s)(Y — 3 3 ) 2 ] =0 , 
the equivalence of which is easily verified by means of the foregoing relations. 
Determination as to Reality . — Art. Nos. 6 and 7. 
6. To fix the ideas suppose that f—g is positive; then in order that the centres of 
the four circles of inversion may be real we must have f-\- 8 .f-\- -/"Ms positive, 
but <7 + A<7+^-<7+02-#+$3 negative ; and this will be the case \if-\-d, 0 2 i /-M s 
are all positive, but g-\-0, g-\-d„ g + Q 3 one of them negative, and the other three 
3 r 2 
