PROFESSOR CAYLEY ON THE BICIRCTJLAR QUARTIC. 
457 
where •>/© is a mere constant ; and we may apply it to the Gaussian transformation, 
a + a! cosT + a" sinT 
cos u= c + ( j cos T + c" sin T* 
b + V cos T -f- b" sin T 
sm a c + c' cos T + c" sin T’ 
where the coefficients a, b, c, a ', b', d, a ", b 1 ", c" are such that identically 
cos»«+sm».-l= (e+e , cos t + c ,, sinl y {cos*T+sin*T-l} 
(cos co b — a) 2 + (sin a \/ g -f- b — /3) 2 — y 2 , that is 
cos 2 ^(/ i +0)4-sin 2 ft>(<7 + #) — 2 a \/ f-j-b cos <o—2fi g+b sin to-\-k 
= 7 1 J, , „ ■ rriN (Gj— G 2 cos 2 T— G 3 sin 2 T). 
(c + c'cosT + c"sin 1) via 3 ' 
30. It is found that G„ G 2 , G 3 are the roots of a cubic equation 
(G +d-Q l )(G+b-b 2 )(G+b-b 3 ), 
which being so, we may assume G^flj — b, G 2 =b 2 — b, G 3 =0 3 — b, or the second condition 
in fact is 
(f+b) cos 2 w + (^+^) sin 2 co— 2a*yf+b cos co— 2/3\A/+$ sina>-|-& 
= ( c+ ^osT + ^.T)» 
and this being so, we find without difficulty the values 
# + 01 
• /+ 0 2 
■/+ 0 3 
b 2 = 
f+K 
• # + 02 
• #+ 0 3 
c 2 _ /+01 
•#+ 0 i 
/-#. 
0 i - 0 2 - 
01 — 0 3 ’ 
#-/• 
01 - 0 2 
. 01 - 03 * 
- 0 i — 0 3 ’ 
# + ®2 
•/+01 
•7 + 03 
b ' 2 =- 
_y+ 0 2 1 
■# + 01 
•# + 03 
/+ 0 2 
• #+ 0 2 
/-#• 
02-01 • 
02 - 0 3 ’ 
#-/• 
02-01 
• 02 - 03 ’ 
- 
• 02 - 03 ’ 
# + 03 
•/+« 1 
•/+0 2 
b " 2 = 
_/+0 3 
•# + 01 
• # + 02 
JI2 /+ 0 3 
•# + 0 3 
/-#• 
03-01 
. 03 - 02 * 
#-/ 
• 03-01 
• 03 - 02 ’ 
- 8 s - i 7 
- 08 — 0 *’ 
(to make these positive the order of ascending magnitude must, however, be not as 
heretofore b 3 , 0 15 b 2 , but b 3 , b 2 , 5,, viz. we must have /+3 1? f-\-b 2 , f+b 3 , g+b x , g + b 2 , 
— (ff+Qs), — ®i—b 2 , b 2 —b 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 
3 T 
MDCCCLXXVII. 
