120 PRINCIPLES OF THE SOLUTION OF EQUATIONS 
$61. Proposition XXII. The auxiliary biquadratic g (~) = 
either has all its roots rational, or has a sub-auxiliary (see $53) of the 
second degree, or is irreducible. 
It will be kept in view that the sub-auxiliaries are, by the manner 
of their formation, irreducible. First, let the series (54), containing 
the roots of the sub-auxiliary ¢1 (z) = O consist of a single term 4). 
Then, by Prop. III., 4; is rational. Therefore, by Prop. XX., all 
the roots of the auxiliary are rational. Next, let the series (54) 
consist of the two terms 4; and 4): By this very hypothesis, the 
auxiliary biquadratic has a quadratic sub-auxiliary. Lastly, let the 
series (54) contain more than two terms. Then it has the three terms 
4,, 49, 43. We have shown that these must be severally equal to 
terms in (64). Neither 4; nor 43 is equal to 4;. They cannot 
both be equal to h® At . Therefore one of them is equal to one of the 
terms a) 4}, & 4°. But in $60 it appeared that, if 4, be equal 
either to a; A or to e Ae all the terms in (64) are roots of the 
irreducible equation of which 4; isa root. The same thing holds 
regarding 43. Therefore, when the series (54) contains more than 
a foneae) the irreducible equation which has 4, for one of its roots 
has the four unequal terms in (64) for roots; that is to say, the 
auxiliary biquadratic is irreducible. 
2 
4 2 
§62. Let OU = 4? 5 Hue =a A? 9 dug = = e, A » OU = hy ae 5 
and, » being any whole number, let S, denote the. sum of the nth 
powers of Hie roots of the equation F lies 0. Then 
S: = 0; 5S, = 10 ia us + w2 Us ) S3 = 15 } S (uy wd) f; 
S, = 20 {z (ui us ) | , + 30 (uj un as us u3) + 120 uy ug ug M4 | 
Ss = 5B § ES (uz) } + 100 4 Sah wg ve) $ + 150 { E (er we ws) 5 
where such an expression as ¥ (w 3) means the sum of all such 
terms as vj UR ; it being understood that, as any one term in the 
circle uw, U2, Us, U3, passes into the next, that next passes into 
its next, wg passing into uw . 
Tue Roots or THE AUXILIARY BIQUADRATIC ALL RATIONAL. 
§63. Any rational values that may be assigned to 4d; , a, e&, and 
hi in 7, taken as in (62), make 7, the root of a rational equation of 
the fifth degree, for they render the values of S,, S2, ete, in $62, 
rational, In fact, S; = 0, 25 Sy = 10 4; (hy + a e ), and so on, 
