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