230 
ON THE BICIRCULAR QUARTIC. 
[667 
or finally 
dA " - vfvk 17 - <* + * -£m ++ ^ 
dY= £m [x - (f+e) = ~J£i [R ' x+a ■ </+e) x] - 
15. We have 
(R’x + a —/+ 6 x) 2 + (R'y + /3 — g + 0 yf 
= R! 2 {x 2 + y 1 ) — 2R' (1 — ax — /3y) 
+ (« -/+ Qx) 2 + (/3-g + 0y) 2 ; 
viz. this is 
= ( a “/+ e x f + (£ - g + 0 y) 2 - r 
= 8 2 , the radius of the generating circle. 
Hence if dS, = VdX 2 + dY 2 , be the element of arc of the bicircular quartic, this 
element being taken to be positive, we have 
ia e'R'Sda) 
do = - , 
Vn V© 
where e denotes a determinate sign, 4- or —, as the case may be. 
16. I stop to consider the geometrical interpretation; introducing dv, the formula 
may be written 
e . R' (x 2 + y 2 ) 8 dv 
~ 7E ’ 
and we have (x 2 + y 2 ) R' = 1 — ax — /3y — Vil, or 
(x 2 + y 2 ) R' _ 1 — ax — fiy 
\/n Vn 
Here -- g f is the perpendicular from the centre of the circle of inversion upon 
VX 2 + y 2 
Vil 
the tangent to the dirigent conic, and — is the half-chord which this perpendicular 
V X 2 + y 2 
forms with the generating circle. Hence -——-- — 1 = (perpendicular — half-chord) 
vil 
-4- half-chord, the numerator being in fact the distance of the element dS (or point 
X, Y) from the centre of inversion: the formula thus is 
d8= + ^ B dv, 
where 8 is the radius of the generating circle, p the distance of the element from 
the centre of the circle of inversion, and c the chord which this distance forms with