122 PRINCIPLES OF THE SOLUTION OF EQUATIONS 
fifth degree ; for they render the values of S;, S2, ete. in §62, 
rational. In fact, S, = 0, 25 S2 = 10{q + q202 — q? d? (p? —¢@°)}, 
and so on. 
THE AUXILIARY BigapRATIC IRREDUCIBLE. 
§66. When the auxiliary biquadratic is irreducible, the unequal 
particular cognate forms of 4 are, by Prop. III., four in number, 
4,, 42, 43, 4g. As explained in §55, because the equation 
¢g (c) = O is irreducible, these terms are severaly identical with 
4,, 62, 63, 64. Hence, putting m = 5, the first two terms in the 
first of the groups (59) may be written in the notation of (37), 
a aa a 
Ay 4g, 42 Ag ; (66) 
and the second and third groups may be written 
$s Ss 5 oS % oS 
And As | Anumeweect sari Ay 4 
(4; 43 , 42 4; , 43 dy 4 499 (67) 
oe ee 
(Avda, ay “ae ae Ay 5” Ag Age 
$67. Proposition XXIV. The roots of the auxiliary biquadratic 
equation g (a) = 0 are of the forms 
A=m+nJf/24+V/8, 44=m—nJY/z+V/5, ) (68) 
A= me + fe J/ 4 =m — 22 — eee 
wherres=p+qVJ % ands; = p—qdJ/2;m™, n,2, p and q 
being rational ; and the surd // s being irreducible. 
By Propositions XIII. and XIX., the terms in (66) are the roots 
of a quadratic. Therefore 4, 4s and dg 43 are the roots of a quad- 
ratic. Suppose if possible that 4; 43 is the root of a quadratic. By 
1 3 
Propositions IX. and XIX., ae =e, 4; . Therefore e At is the 
root of a quadratic. From this it follows (Prop. III.) that there are 
not more than two unequal terms in the series, 
5 4 5 5 44 
ey abe e Ao, é3 Ap 4 Algy (69) 
But suppose if possible that e At = es ae Then, ¢ being one of the 
4 4 
fifth roots of unity, te, 4 : = e, 43 But, by Propositions IX. and 
1 4 4 1 
XIX., 4) = hy AP. Therefore, te; 4p = ¢2 hy 4; 47 . There 
