PROFESSOR CAYLEY ON THE BICIRCULAR QUARTIC. 
459 
(0+/) cos 2 a>+(0+Z sin 2 <y— 2a ^Z^+Zcos co — 20^/0 +g sin a-\-k 
l 
(c + c' cosT + c" sin T) 2 
{(0X— 0)— (0 2 — 0) cos‘ 2 T— (0 3 — 0) sin 2 l+ 
(0, - 0) - (0 2 - 0) cos 2 T - (0 3 - 0) sin 2 T 
Tacota + L mv -c)* * 0+/) cos 2 <y+(0+^) cos »-2/3 x/0+<7 sin »+Jc}, 
giving the four sets each of six equations 
a 2 +0 2 — c 2 = — 1, 
d 2 +b' 2 —d a = + l, 
a"* +#»-(?' 2 =+l, 
a'a"+W"-c , c"=0, 
-\-b"b—d'c =0, 
ad -\-bV —cd =0, 
a 2 +a' 2 + a" a = + l, 
5 2 +5' 2 +6" 2 =+l, 
C 2 +C' 2 + ^2 =r _ 1? 
-Jc+J'c'+&"c"=0, 
— ca + cW + c"«" = 0, 
-ab + db'+a"b"=0, 
( b-\-f)a 2 +(0+</)^ 2 — 2a>/0+Z ac — ^^/b+gbc 
{b+f)d 2 +(b+g)b' 2 -2a*/Q+f dd -2fa/$+gb'd 
(0+/K* + (5 + Z5" 2 - ZoLy/fi+f'd'd 1 -2(ds/Q+jb"d' 
-\-kc 2 = 0,-0, 
+£c' 2 =-0 2 +0, 
+£c" 2 =— 0 3 +0, 
(0+/)a , a"+(0+Z0 , 0' , -ax/0+/(«V , +«V)-f3 x /0+^ (5'c"+&V)+&V'=0, 
(0+/K« +(0+#"i -aLs/J+f(d’c+ad’ )-(3 x/0+^ (0"c +0c" )+&c"c =0, 
(0+/)aa' +(0+Z0^ — a^/0+/ (ac'+a'c) — /3 a/$+</ (0^ +^<?) -\-Jccd =0, 
(3 1 _0)a 2 _(5 2 -0)a' 2 _(03_0y 2 = 0+/, or say (0 1 +/)« 2 -(0 2 +/y 2 -(0 3 +/>" 2 = O, 
(0x — 0) 0 2 — (0 2 — 0)0' 2 — (0 3 — 0) 5" 2 = 0 + <7, „ (0 1 +# 2 -(0 2 + #' 2 -(03+^)^ 2 = O, 
(0 1 -a) c 2 -(0 2 _0y 2 _(3 3 -~0) c " 2 ^: / 5:, „ 0 lC 2 - 0 2 c' 2 -0 3 c" 2 =A+0, 
-(0 1 -0)0c+(0 2 -0)0V+(0 3 -0)0V'=-^ N /0+Z, 
— (0j — 0)<?a +- (0 2 — Q)dd + (0 3 — b)d'a" = -ajb+f\ 
(0 1 — b)ab — (0 2 — Q)dV — (0 3 — Q)a"b” = 0 ; 
all which formulae are in fact satisfied by the foregoing values of the expressions 
a 2 , b 2 , a' 2 , &c. 
