458 
PROFESSOR CATLET ON THE BICIRCULAR QUARTIC. 
f±\ / ®o®, 
n _ 6~ — A J \r 
/+«* 
~f~9 • 9 2~ S 3 
3 “2 
i'5"=- 
-/ 
A r 
■ K-hV 
c'c"= 
h s 
j-g 
c " c= \f 
®3®1 
2 -V 
aa / T+ 6 ? / - «i© 2 
AJV ^ . / ~ 
bb ~9-f-K-^\l 
cd = s^j; v ’’ j 
and further 
«/> = 
dV = 
a"b" 
1 / /w >n 7 r — / +9 i‘ , /9 + ^1-9 + ^-9 + h ca—— + ■■ -A / Z+W+^-H 
.t 3 -S lA -ty-®^> b 6 a -tiA-kY F7 ’ fls-fil-fil-flaV f-g 
” ’ c,ft — e 3 -^ 1 j 1 2 _5 2 \/ ” I i 
” ’ C ^ ,,= A/ 
-L.L-L*J 
f-9-0 1-M2-H 
, 6V= 
/+** . / 
-a 2 .fl 2 -fl 3 V 
-^•^2— d 3* 0 3 — 
and also 
\/ „ ,b"c"— Q \f 
jut \yu— 2 9+K+h / 9+K-f+^-f+h ,J n " — ^/+^2 + g 3 / .t+^i ■ g + Qz- g+&a 
bc+bd ~ e 2 -<?3 V ,-/.* 8 -vfl,-« r 2 > s 2 -fl 3 V 7 ^. 
w + s ' c =2 w V-f $£Si£i ' *+* = 1 j^t 
32. These values, in fact, satisfy the several relations which exist between the nine 
coefficients, viz. the original expressions of cos a, sin a/, in terms of cos T, sin T give 
conversely expressions of cos T, sin T in terms of cos a>, sin <y, the two sets being 
a + a' eos T + «" sin T „ a' cos a> 4- # sin co — c f 
COS a>— - — , _ - „m . -il rrn COS I 
'c + e'cos T + c" sin T’ 
6 + 6' cos T 4- b" sin T 
sm u c + c' cos T + c" sin T’ 
and we have then the relations 
sin T = — 
a cos as + b sin co — c’ 
a" cos 03 + b" sin 03 — d 1 
a cos co + 6 sin 03 — c ’ 
cos*.+sm*« ^ = { c + o' cos T+ 7' sin T) a (™s 2 T+sin 3 T-l), 
cosT+sin 2 T- l= r c - cosa , + Lin«,-o) 2 (cos ! *+sin 2 *- 1), 
