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»