135 
741] ON A THEOREM OE ABEL’S RELATING TO A QUINTIC EQUATION. 
viz. these are the imaginary fifth roots of unity, or roots r, r 2 , r*, r 3 of the quartic 
equation x* + a? + oc 2 + x +1 = 0 \ which equation, as is well known, has the group 
y^2^>4^»3 ^'»3,^»^»2^»4 
Reverting to Abel’s expression for x, and writing this for a moment in the 
form 
x = c+p +s + r+q, 
the quintic equation in x is 
0 = (x — cf 
4- (x — c) 3 . — 5 (pr + qs) 
+ (x — cf. — 5 (p 2 s + (fp + r'-q + s 2 r) 
+ (x — c) . — 5 (p 3 q + q 3 r 4- r 3 s 4- s"p) + 5 (p 2 r 2 + q 2 s 2 ) - opqrs 
+ (x — cf.— (p° + q 5 + r 5 + s r °) 
4- 5 (p' J rs 4- q 3 sp + r'pq + s 3 qr) 
— 5 (p 2 q 2 r + q 2 r 2 s + r 2 s 2 p + s 2 p 2 q). 
If we substitute herein for p, q, r, s their values, then, altering the order of the 
terms, the final result is found to be 
0 = (x — c) 5 
4- (x — c) 3 . - 5 (ii 2 4- AjA-j) aa 1 a 2 a ;i 
+ (x — cf. — 5 (A 2 A 1 a.,a- i + A 1 2 A 2 a 3 a + A 2 2 A 3 aa 1 + A s 2 Aa 2 a 2 ) aa^iM, 
+ (x — c) . — 5 {A 3 Apb 1 a^a i 4- AfAa 2 afa 4- A.fA^d 2 ^ + A^A.uafa.,) aa^aau, 
4- 5 (A 2 A.i 2 4- A 2 A 2 — AiA.A ;i A) (aa x a.nff 
+ (x — cf. — (A^aquaf + Afa 2 afaf + Aja/Aa-f 4- Afaafa.f) 
4- 5 (A 3 A a A 2 a 2 a y 4- AfA 2 Apb^a + AsA-.Aa^ + A 3 3 AA 1 u 1 a 2 ) (aa^ua..)' 1 
- 5 {A 2 A.?A. i a l a 2 + A{-A 2 A.pM, + A£AfAa 3 a + A 3 2 A 2 2 A 1 aa 1 ) (acqa^) 2 ; 
viz. considering herein A, A 1; A 2 , A ; . as standing for their values 
K -f K'a + K"a, + K'"aa 2 , &c. 
respectively, each coefficient is a function of a, a 1} a 2 > a 3> which is unaltered by the 
cyclical change of these values and therefore is a rational function of