PROFESSOR CAYLEY ON THE BICIRCULAR QUARTIC. 
447 
We have 
that is 
xX-\-yY —ax-\-fiy-\-{o(? +?/ 2 )R' 
= 1- n /D; 
1 — xK—yY 
and consequently the foregoing^ expressions of dX., dY become 
dX=~~{( S +6) ! ,(xX+ ;/ Y-l)+4-( g +S)yX + (f+^Y)' s 
dY= ^k { (/+«W1 -*X-yY)+y(- (<?+%X+(/+6>Y) } 
= W7S^ + ^-(^ + ^ +(?+ 9 )/) x f> 
or finally 
dX= 
R'^w ( 
V© 
dY= 
-R ! dm , 
*/©viT 
X-(/+%}, 
=WVH»+' 3 -^+W’ 
15. We have 
( R ' a !+ a -/+ fl ^) 2 +( R ^+/ 3-^+^) 2 
= ~R'\a? -\-y 2 ) — 2R'( 1 — ax — fly) 
+{ a -FrQx) 2 +{P-¥+hY’ 
viz. this is 
=(u-f+Qxy+(P-9+ Q y) 2 -r 2 ’ 
=l 2 , the radius of the generating circle. 
Hence if $3, =\/ dX 2 +dY 2 , be the element of arc of the bicircular quartic, this ele- 
ment being taken to be positive, we have 
£ 'R'8<fo 
d ^~ Vxi V©’ 
where s' denotes a determinate sign, + or — , as the case may be. 
