Full text: The collected mathematical papers of Arthur Cayley, Sc.D., F.R.S., late sadlerian professor of pure mathematics in the University of Cambridge (Vol. 11)

EQUATION. 
517 
and, the periods being known, the symmetrical functions of the several terms of each 
period are rationally determined in terms of the periods, thus 
■y» —- jp^ ^ V • V • T* 4" V 2 . = JPo, T • V 2 T^ = 1. 
The theory was further developed by Lagrange (1808), who, applying his general 
process to the equation in question, x n ~ l + x n ~ 2 +...+«+1=0, the roots a, b, c,... being 
the several powers of r, the indices in geometrical progression as above, showed that 
the function (a + o>5 +&) 2 c+was in this case a given function of w with integer 
coefficients. Reverting to the before-mentioned particular equation « 4 + « 3 + « 2 + x +1 = 0, 
it is very interesting to compare the process of solution with that for the solution 
of the general quartic the roots whereof are a, b, c, d. 
Take a, a root of the equation eo 4 —1 = 0 (whence &> is =1,-1, i, or —i, at 
pleasure), and consider the expression 
(a + cob + co 2 c + co 3 dy. 
The developed value of this is 
= a 4 + b 4 + c 4 + d 4 + 6 (a 2 c 2 + b 2 d 2 ) + 12 (a 2 bd + b 2 ca + c 2 db + d 2 ac) 
+ to {4 (a?b + b 3 c + c 3 d + d 3 a) + 12 (a 2 cd + b 2 da + c 2 ab + d 2 bc)} 
+ (o 2 {6 {a 2 b 2 + b 2 c 2 + c 2 d 2 + d 2 a 2 ) + 4 (a 3 c + b 3 d + c 3 a + d 3 b) + 24abed) 
+ to 3 {4 (a 3 d + b 3 a + c 3 b + d 3 c) +12 (a 2 bc + b 2 cd + c 2 da + d 2 ab)} ; 
that is, this is a 6-valued function of a, b, c, d, the root of a sextic (which is, in 
fact, solvable by radicals; but this is not here material). 
If, however, a, b, c, d denote the roots r, r 2 , r 4 , r 3 of the special equation, then 
the expression becomes 
r 4 + r 3 + r + r 2 + 6 (1 + 1) + 12 (r 2 + r 4 + r 3 + r) 
+ to {4(1 + 1 +1 + 1) + 12 (r 4 + r 3 + r + r 2 )} 
+ ft) 2 {6 (r + r 2 + r 4 + r 3 ) + 4 (r 2 + r 4 + r 3 + r )} 
+ a) 3 {4 (r + r 2 + r 4 + r 3 ) + 12 (r 3 + r + r 2 + r 4 )]; 
viz. this is 
= — 1 + 4 to + 14 eo 2 — 16&) 3 , 
a completely determined value. That is, we have 
(r + <wr 2 + &) 2 r 4 + g) 3 r 3 ) 4 = — 1 + 4ft) + 14ft) 2 — 16ft) 3 , 
which result contains the solution of the equation. If &> = 1, we have (r+ r 2 + r 4 + r 3 ) 4 = 1, 
which is right; if w = — 1, then (r + r 4 — r 2 — ?- 3 ) 4 = 25 ; if &) = i, then we have 
[r — r 4 + f (r 2 — «3)} 4 = _ 15 + 20i; and if &) = — i, then [r — r 4 — i (r 2 — r^)} 4 = — 15 — 20f; the 
solution may be completed without difficulty. 
The result is perfectly general, thus:—n being a prime number, r a root of the 
equation x n ~ 4 + x n ~ 2 + ...+« + 1=0, w a root of a> n ~ l —1 = 0, and g a prime root of 
g 1l ~ 1 = 1 (mod. n), then 
(r + u>r g + ... + u) n ~ 2 r gn ') n_1
	        
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