ON THE BICIRCULAR QUARTIC. 
or say the two values are 
1 — ax — /3y + Vil 
1 — ax — {3y — Vil 
X- + y~ ' oc 2 + y 
to preserve the generality it is proper to consider Vil as denoting a determinate 
value (the positive or the negative one, as the case may be) of the radical. 
11. Considering the root R 1 , we have X = a + R'x, Y = (3+R'y; from these equa 
tions we obtain 
dX = R'dx + x dR', 
dY = R'dy + y dR'. 
But from the equation for R' we have 
[R' (os 2 + y 2 ) — (1 — ax — ¡3y)~\ dR' + R' 2 (x dx 4- y dy) + R' (a dx + ft dy) — 0, 
- Vil dR' + R' (Xdx + Ydy) = 0 ; 
that is, 
whence 
R'r 
dX = R'dx + (Xdx + Ydy), 
vil 
dY = R'dy + R l (Xdx + Ydy). 
vil 
12. The differentials dx, dy can be expressed in terms of a single differential day, 
viz. writing 
x — 
COS 0) 
Sin ft) 
and 
then we have 
V/+0’ y \lg + d’ 
® = (f+0)(g+e\ 
j 9 "h @ ? j / + 0 7 
dx — — y dw, dy — x do). 
Vb j Vb 
It is to be observed that, when the dirigent conic is an ellipse, w is a real 
angle, and B is positive (whence also VB is real and positive); but when the dirigent 
conic is a hyperbola, w is imaginary, and B is negative; we have, however, in either 
case 
da? + df = (/±i)^+(?±g)V dm , t 
and we may therefore write 
dco ds 
VB (f + 6) 2 x 2 y- (g + 6) 2 y 2 ' 
where V(/+ 6) 2 x 2 + (g + d) 2 y 2 is positive; ds is the increment of arc on the conic 
(/+ 6)x 2 + (g -f 6) y 2 = 1, this arc being measured in a determinate sense, and therefore 
ds being positive or negative as the case may be : has thus a real positive or 
negative value, even when « is imaginary, and it is convenient to retain it in the 
formulae.