242 
ON THE BICIRCULAR QUARTIC. 
[667 
giving the four sets each of six equations 
a 2 +b 2 -c 2 =-1, 
a' 2 + 6' 2 - c' 2 = 4 1, 
a "2 + y , 2_ c " 2= + i ( 
- a 2 4- a' 2 + a" 2 = 4 1, 
- 6 2 4&' 2 4 6" 2 =4l, 
- c 2 4 c' 2 4c" 2 =-l, 
(0 4 /) a 2 4(0 4 0)6 2 -2a/61+fac 
{0+f)a' 2 4(0 4 0)6' 2 — 2 a /0 4/a'c' 
a'a" 4 b'b" - c'c" = 0, 
a"a 4 6"6 - c"c = 0, 
aa' 4 66' — cc' = 0, 
-6c 46V +6"c" = 0, 
— ca 4- c'a' 4- c"a" = 0, 
— a6 4 a'b' 4- a"6" = 0, 
— 2/3 /0 4- g be 
— 2/3 /0 + g b'c 
— 2/3 /0"40 b"c" 
4 kc 2 = #i 4 0, 
4&c' 2 = — # 2 4#, 
4" kc” 2 = — 0 3 40, 
(0 + /) a" 2 4- (0 4 g) 6'' 2 - 2 a \/d 4 /a"c 
(0 4/) a'a" + (6+g) b'b" - a /04/(a'c" 4 a"c') - .3 /040 {b'c" 4 6"c') + ke'e" = 0, 
(0 4 /) a"a + (0 4 g) b"b — a /0 4 f(a"c 4 ac") — /8/0 40 (b"c 4 be" ) 4 kc"c = 0, 
(0 4 f) aa' 4 {0 4 g) 66' — a /0 4/(ac' 4a'c)— /3/0 4 0 (6c' 4 6'c ) 4 kcc' = 0, 
(0! — 0) a 2 — (0 2 — 0) a' 2 — (0 3 - 0) a," 2 =0 4/ or say (0j 4/) a 2 — (0., 4/) a' 2 —(0 3 4/)a" 2 =0, 
(^ - 0) 6 2 - (0 2 - 0) 6' 2 - (03 - 0) 6" 2 = 040, „ (0! 40) 6 2 - (0 2 4 g) 6' 2 —(0 3 40 )6" 2 =0, 
(0 1 -0)c 2 -(0 2 -0)c' 2 -(0 3 -0)c" 2 = ^, „ 0jC 2 - 0 2 c' 2 - 0 3 c" 2 =¿40, 
- (0! - 0) 6c 4 (0 2 - 0) 6'c' 4 (03 - 0) 6"c" = - /3/0 4 0, 
- (0 X - 0) ca4 (0 2 - 0) c'a' 4 (0 8 - 0) c"a"= - a /04/ 
- (0, - 0) a64 (0 2 - 0) a'b' 4 (0 3 -0) a"6"= 0 ; 
all which formulae are in fact satisfied by the foregoing values of the expressions 
a 2 , 6 2 , a' 2 , &c. 
33. We then have 
da) =■ 
dT 
c 4 c' cos T 4 c" sin T ’ 
the radical which multiplies dw being 
c 4 c' cos T 4 c" sin T ^ “ 6,2 cos ‘ 2 T ~ 6 * sin2 T > 
the differential becomes 
dT /0 X — 0 2 cos 2 T — 0 3 sin 2 T 
that is, 
cos 2 to sin 2 to 
./4 0 04 0 
4 c' cos T 4 c" sin T) 2 /© 
d ; T /0i - 0, cos 2 T -0 3 sin 2 T 
^ (a 4 a' cos i 7 4 a" sin /) 2 4 ^ (6 4 6' cos T4 6"sin 7 1 ) 2 ! /© 
1 P . /1 \U/ T" U/ Vy'J’O _x p L(/ oxn x / t 7» 
(/+ 0 0 4 0 
The denominator could, of course, be reduced to the form (*$1, cos T, sin T) 2 -, but 
the actual form seems preferable, inasmuch as it puts in evidence the linear factors 
(a 4 a' cos T 4 a" sin T) 4 . (646' cos T 4 6" sin T), 
//4 0 /0 4 0 
and there seems to be no advantage in further reducing the integral.