667] 
ON THE BICIRCULAR QUARTIC. 
241 
and farther 
ab 
alb' = 
a"b" = 
1 fi»iafiii i c - f+0i /g + 0i-g + 02-g + 0 3 
f-g.ds-e^-e, UlUM ’ 6C - 0 t -0 l .0 l -0 t V g-f 
v ôv = / /. ^ . / 
d x -e>.o,-ey 
/ + 0 3 / 
v ,, , U U „ s, s, ^ ^ „ , 
f-g.0 1 -d».6„-6 s 
-1 
/-0.0J-0S.03-0! 
ca = 
0 2 - 0 3 .03-0, 
+ 02 
f-g 
g + 0i // + 0i •/+ 02 •/+ 0 3 
03-0!. 01-02 V /-0 
0 + 02 / 
0 1 -0 2 .0 2 -0 3 V 
? + 
0,-03.03-0, 
and also 
/)V" I AV — ^0 + 02+03 / 0+0i./ + 0 2 -/+ 03 / //. // /_2/+ 0 2 +03 / /+01-0+02-0+03 
+ ö„-ö, v g-fA-0,A-0f 0,-0, V f-gA-e,A-0f 
+ 0,-0, w g -fA-e,A-»,' V f- g .0,-0,a-0/ 
7 / A-h'r - ^0 + ^1+^2 / 0+ 03-/ + 01-/+02 / , / _ 2 /+ 0!+ 02 / /+03-0+ 01 -0+ 02 
+ - 0,-0, V g-fA-0.A-0f 8,-0, V f-g.0,-0,A-0,' 
32. These values, in fact, satisfy the several relations which exist between the 
nine coefficients ; viz. the original expressions of cos to, sin <w, in terms of cos T, sin T 
give conversely expressions of cos T, sin T in terms of cos a), sin &), the two sets being 
a + a' cos T 4- a” sin T rp a' cos eo + 6' sin w — d 
c + c cos T + c sin _/ a cos îo + b sin co — c 
b + b' cos T + 6" sin T . ^ a" cos w + b" sin w — c" 
Sm a) — T,—■ 7n , sin 1 — j—; : 
c + c cos T + c sm T a cos w + b sm <u - c 
and we have then the relations 
cos 2 « + sin 2 (o-l= ( c + c > cos T + c " gin Tf ^ C ° S2 T + Sm2 T ~ 
COS 2 T + Sin 2 T— 1 = 7 V 2 (COS 2 ft) + Sin 2 ft) - 1), 
(a cos co + 6 sm <w — cf 
(0 + /) cos 2 « + (0 + 0) sin 2 &) - 2a V0 + /cos to - 2/3 V0 + g sin « + & 
{(0, - 0) - (0 2 - 0) cos 2 T - (0 3 - 0) sin 2 1 7 }, 
(c + c' cos T + c" sin Tf 
(0, - 0) - (0 2 - 0) cos 2 T - (03 - 0) sin 2 T 
L_ !(0 + f) cos 2 ft) + (0 + 0) sin 2 ft) - 2aV0 + /cos to - 2/3 /0 +# sin « + k], 
(a cos co + b sin to — c) 21 J 
C. X. 31