402 
[777 
777. 
A SOLVABLE CASE OF THE QUINTIC EQUATION. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xvm. (1882), 
pp. 154—157.] 
The roots of the general quintic equation 
(a, h, c, d, e, l) 5 = 0 
may be taken to be 
--+ B+ C+ D+ E 
- „ + CO* „ + CO 3 „ + (0 2 „ + 03 „ 
- „ + CO 3 „ + CO „ + CO* „ + (O 2 „ 
~ „ + CO 2 „ + CO* „ + CO „ + CO 3 „ 
- „ + (O „ + CO 2 „ + CD 3 „ + CO* „ , 
where co is an imaginary fifth root of unity; and if one of the four functions B, 
C, D, E is = 0, say if E = 0 (this implies of course a single relation between the 
coefficients), then the equation is solvable. 
Writing x = f ^, we have 
(a, b, e, d, l) = = (a'. 0, o', d', e'.f'H 1)», 
where 
a' = a, 
ac' = ac — h 2 , 
a 2 dl — a 2 d — Babe + 2b 3 , 
add = a 3 e — 4ta 2 bd + 6ab 2 c — 3b*, 
a*f = a*f — 5ci s be + 10a& 2 d — 10a& 2 c + 46 5 , 
and the roots of the new equation 
<«', 0, o', d', 1) 5 = 0