126 PRINCIPLES OF THE SOLUTION OF EQUATIONS 
K, K', K” and K” being rational. But the terms Ai, Ag, Ag, 
Ay circulate with 4; , 42, 44, 43. . Therefore 
Ag=K4+ K’' 4+ K’ 4, + K’” do 43, 
Ay= K+ K' a+ K’ A+ K™ Ay Aas 
Azs=K4+ K' 4,+ K” 46+ K" Ao 43, 
These are Abel’s values. 
§72. Keeping in view the values of 4), 42, etc., in (67), and also 
thatz = 1+e,ands=hz + h/ 2, any rational values that may 
be assigned to m, n, e, h, K, K’, K” and K'” make r;, as presented 
in (74), the root of an equation of the fifth degree. For, any rational 
values of m, 7, etc., make the values of S; , S2, etc., in $62, rational. 
§73. It may be noted that, not only is the expression for 7; in (74) 
the root of a quintic equation whose auxiliary biquadratie is irre- 
ducible, but on the understanding that the surds ./ s and ,/ zin 
4, may be reducible, the expression for 7; in (74) contains the roots 
both of all equations of the fifth degree whose auxiliary biquadratics 
have their roots rational, and of all that have quadratic sub- 
auxiliaries. It is unecessary to offer proof of this. 
§74. The equation # — 1003 + 5a? + 10% + 1 = O is an 
example of a solvable quintic with its auxiliary biquadratic irre- 
ducible. One of its roots is 
3 4 
ae ww” aE wtw? ; 
w being a primitive fifth root of unity. _It is obvious that this root 
satisfies all the conditions that have been pouited out in the preceding 
analysis as necessary. A root of an equation of the seventh degree 
of the same character is 
oreo 
1 
w® + ww 
€ 
3 4 5 6 
Gi of a, Gon) 
+ vtot + wv? + wtw? + v8o7, 
Lo 
1 
wo! + wo 
w being a primitive seventh root of unity. The general form under 
which these instances fall can readily be found. Take the cycle that 
contains all the primitive (m’) roots of unity, 
0, 08, 0, ete. (75) 
m being prime. The number of terms in the cycle is (m — 1)?. 
Let 6; be the (m+ 1)™ term in the cycle (75), 02 the (2m + pie 
term, and so on. Then the root of an equaticn of the m degree, 
including the instances above given, is 
= 
m= (64+ 0-1)+ (1 +0-) + .... + 6, 34 
2 2 
