PEOFESSOE CAYLEY ON THE BICIECTTLAE QTJAETIC. 
445 
the derived equation is 
(X— «X/+tyto+(Y— P)(g+*)dy=0 ; 
and from these two equations, together with the equation in (x, y) and its derivative, 
we find X — a=R#, Y — /3=R/y; from these last equations, and the equations 
R=^(X 2 + Y 2 — k), 1, eliminating x, y, R, we have 
(/+d)(X-af+( S r+«)(T-g) s =E> > 
that is 
(X a +Y ! -^-4[(/ +1 ))(X- a ) ! +( ? +^)(Y-OT=0, 
the required equation of the bicircular quartic. 
10. We have thus X— a=R#, Y— /3=Ry, as the equations which serve to determine 
the bicircular quartic: if from these equations, together with R=-^(X 2 +Y 2 — 1c), we 
eliminate X, Y, we have R expressed as a function of x, y; and thence also X, Y 
expressed in terms of x, y ; that is in effect the coordinates X, Y of a point of the 
bicircular quartic expressed as functions of a single variable parameter. The process 
gives 2R+A:=(a+R^) 2 + (/3 + R?/) 2 , viz. this is 
R 2 (# 2 +^ 2 ) — 2(1 — ax -/ty)R+ y 2 = 0, 
or putting for shortness 
O,=(l — oix—(5yy~y 2 (x 2 +y 2 ), 
this is 
R 
l —ctx— fiy + \Afit 
x 2 +y 2 
or say the two values are 
-o 1 — ax — /3y + V'Xl 
x 2 +y 2 
, R': 
1 —ux—fiy— */ £1 . 
x 2 + y 2 
to preserve the generality it is proper to consider ^/Q as denoting a determinate value 
(the positive or the negative one, as the case may be) of the radical. 
11. Considering the root R' we have X=a + R'^, Y =/3+R , yn and from these equations 
we obtain 
dK.=BIdx-\-xdBI, 
dY=B!dy+ydR'; 
but from the equation for R' we have 
[_W(x 2 -\-y 2 ) — (1 —ax— fiyJjd'R ! + R ,2 (xdx-\-ydy) + ld!(a.dx-\- (5dy) =0, 
that is 
-s/Q. dB! + 'B!(Xdx+Ydy)=0, 
whence 
<D£=E'fo+^ (Xdx+Ydy), 
dY=U'd!,+^(Xdx+Ydy). 
