ON THE BICIRCULAR QUARTIC.
[667
cos 2 w (f+ 0) + sin 2 a) (g + 0) — 2a ff + 0 cos <w — 2/3 fg + 0 sin w + Jc,
= (c + c' cos T+c" sin Tf ( Gl ~ G * C0S “ T ~° 3 Sm3
30. It is found that G 1} Cr 2 , Cr, are the roots of a cubic equation
(G + e-ejiG+e-ejiG + e-ej,
which being so, we may assume G l = 6 l — 0, G 2 = 0 2 — 0, G 3 = 0 3 — 0; the second condition,
in fact, then is
(/ + 0) cos 2 a) + (g + 0) sin 2 a) — 2a ff+ 0 cos w — 2/3 fg + 0 sin w + Jc
= (c + c'cos T+c" sin Tf ^ ~ 6 ~ ~ 6 ^ C0S2 T ~^ 3 ~ ^ sin2 ;
and this being so, we find without difficulty the values
2 g + 0i -f + 02 •f + 03
f-g. 01- 02.01 - 03’
6 2 =
f+ 0i ■ g + 02 • g -t- 03
g-f.6,-0,.6,-0,'
2 _ /+ 0i-g + 0i
0i-02.0i-03
/ 2 _ g + 02-f +0i •/+ 03
b'* =
f + 02 ■ g + 0i • g + 03
g -/• 02 - 0i. 0 2 - 03 ’
¿2 _ / + 02.g+02
02 — 01.02 — 0 3
„2 g + 03 • f+ 01-f+ 02
6" 2 =
f + 03 • g + 0i • g + 02
g-f-6,-0, .0,-0,’
'/ 2 f+03-g + 0s
0,-0,. 0,-0,
To make these positive, the order of ascending magnitude must, however, be not as
heretofore 0 3 , 6 1 , 0 2 , but 0 3 , 0 2 , 0 1} viz. we must have f+0 ly f+0 3 , f+03, g + 0\,
g + 02, ~(g + 0 3 ), 01-03, 01- 02, 02-0-3 all positive.
31. The above are the values of the squares of the coefficients; we must have
definite relations between the signs of the products aa, bb', ab, &c., viz. we may have
aa =
f+ 0i
/-g.02 - 03 \ 03 - 01-01 - 02’
VU’ = # + ft . /
Q — f. 02— 02 V
a a
1
02-03
b"b =
c c —
f+02 /_ - ©3©1 _
f-g.03-01 V 01-02.0,-03’
g + 02 /
- e x V
g-f-0 3
07=0, \J
, f+02 / “ ©1©2
~f-g.0,-0,^ 6,-6,.6,-6,'
/ _ g + 02 /
~g-f .0i-02 V
bb
cc =
01 - 02