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