667] 
ON THE BICIRCULAR QUARTIC. 
229 
13. It may further be noticed that, if v denote the inclination to the axis of x 
of the tangent to the dirigent conic at the point *Jf\-0 cos &>, V#+ 0 sin co, where 
v is Casey’s 0, then 
COS V Sin V , tt / /• /i\ o / /iv • o 
x = -¡=, y = —, where U ={/ + 0) cos 2 v + (g + 0) sin- v, 
viz. we have 
Vi7’ * Vi7 
COS CO _ cos v sin CO sin u 
V/+ 0 U ’ \/g + 0 U 
giving, as is easily verified, ^; we have therefore 
do) 
dv 
or 
(a; 2 + y 1 ) V(S) v (x 2 + y 2 ) 
^|=(*»+y’)d U , 
— dv, 
which is another interpretation of 
do) 
V© ' 
14. Substituting for dx, dy their values, the formulae become 
dX = j- (g + 0) y + -JL (- (g + 0)yX + (/+ 0) ®F)j dw, 
dY = Jg | (/+ 0) x + (- (g + 0) yX + (/+ 0) æF)j dœ. 
We have 
xX + y Y= ax 4- /% + (x 2 4- y 2 ) R' 
= 1 - Vn, 
that is, 
1 — aX — yY 
7yT~~ ; 
and consequently the foregoing expressions of dX, cZF become 
dX = ^°-\(g + 6)y(xX + yY-l)+x{-(g + <l)yX + (f+e)xY)\ 
<iF = ((/+0)«(1 - ¡rX-yFl + y (-(ÿ + 9)yX + (/+ e)xY)) 
v© Vi2 
= r^= K/+ (9 > « - ((/+ °) æ *+(y + e ) y s ) z î»