442 
PROFESSOR CAYLEY ON THE BICIRCULAR QUARTIC. 
I remark that in the various formulae / , g, 6, 0 15 d 2 , 6 3 are constants which enter only 
in the combinations/^,/— g, 6,-6, d 2 —Q, 0 3 —d, that X, Y are taken as current coor- 
dinates, and these letters, or the same letters with suffixes, are taken as coordinates of a 
point or points on the bicircular quartic; the letters ( x,y ), {x x ,y x ), {x 2 ,y 2 ), (# 3 *^ 3 ) are 
used throughout as variable parameters, viz. we have 
{f+*)*+{g+ 6 )tf= 1 . 
{f+0i)*+{g+6)£= 1 . 
(/-Ma) #2 + (#+^2) y!=l> 
(/ + 4 ») 4 ») ^= 1 ; 
cos w sin co 
are functions of a single parameter &>, and similarly (x x , y/ 
so that x,y= 
*/f+ fl’ Vg + tf 
(x 2 ,y 2 ), (x 3 ,y 3 ) are functions of the parameters cy„ cy 2 , u 3 respectively; we sometimes use 
these or similar expressions of ( x,y ) &c. as trigonometrical functions of a single para- 
meter, but more frequently retain the pair of quantities, considered as connected by an 
equation as above, and so as equivalent to a single variable parameter. 
Formulae for the fourfold generation of the Bicircular Quartic. — Art. Nos. 1 to 5. 
1. We have four systems of a dirigent conic and circle of inversion, each giving 
rise to the same bicircular quartic : viz. the bicircular quartic is the envelope of a gene- 
rating circle, having its centre on a dirigent conic, and cutting at right angles the 
corresponding circle of inversion; or, what is the same thing, it is the locus of the 
extremities of a chord of the generating circle, which chord passes through the centre 
of the circle of inversion, and cuts at right angles the tangent (at the centre of the 
generating circle) to the dirigent conic; the two extremities of the chord are thus 
inverse points in regard to the circle of inversion. The four systems are represented by 
letters without suffixes, or with the suffixes 1, 2, 3 respectively, and we say that the 
system, or mode of generation, is 0, 1, 2, or 3 accordingly. 
2. The dirigent conics are confocal, and their squared semiaxes may therefore be 
represented by f-\-6, g + 0, /-Mi, # + 0 1 , /+0 2 , g+^ /+0 3 , g-\-0 3 (which are in fact 
functions of the five quantities /-M, / — g , $1 — 0 , 0 2 — 0» 0 3 — 0) ; and we can in terms of 
these data express the equations as well of the dirigent conics as of the circles of 
inversion ; viz. taking X, Y as current coordinates, the equations are 
(X-«) 2 +(Y-^) 2 - y 2 =0, or X 2 +Y 2 -2« X— 2/3 Y + k =0, 
( X -«i ) 2 +(Y-^) 2 -y?=0, or X 2 +Y 2 -2 ai X-2^Y+^ = 0, 
,+^y=l. (X-«,)>+(Y-ft)>-,$=0, or X 2 +Y a -2 aj X-2/3 2 Y+l: ! =0, 
X 9 Y 2 
