OF THE HIGHER DEGREES, WITH APPLICATIONS. 123 
fore, by §8, ex = 0. Therefore one of the roots of the auxiliary 
biquadratic is zero; which because the auxiliary biquadratic js 
assumed to be irreducible, is impossible. Therefore et At and es Ms 
are unequal. In the same way all the terms in (69) can be shown to 
be unequal ; which, because it has been proved that there are not more 
than two unequal terms in (69), is impossible. Therefore 4; 43 is not 
the root of a quadratic equation. Therefore the product of two of the 
roots, 4; and 44, of the auxiliary biquadratic is the root of a quad- 
ratic equation, while the product of a different pair, 4; and 43, is not 
the root of a quadratic. But the only torms which the roots of an 
irreducible biquadratic can assume consistently with these conditions 
are those given in (68). 
§68. Proposition XXV. The surd ,/ s; can have its value ex- 
pressed in terms of ,/s and 4/2. 
By Propositions XIIJ. and XTX, the terms of the first of the groups 
(67) are the roots of a biquadratic equation. Therefore their fifth 
powers 
Ai 43, dz 4, 43 4g, AE ds, (70) 
are the roots of a biquadratic. From the values of 4;, 42, 43 and 
Ay in (68), the values of the terms.in (70) may be expressed as 
follows : 
Ads = FEA VY 2+ (t+ hs) Vs ‘| 
4(%44+ 6sV2 Yat (h+AivadsV/ 43, 
44=F—-RhY2+(—-BVYD)V 4% 
— (Mm — Fs V2 8 —(Fe—-MVe2V8V 1; 
4a=F-A Vt (h—- VD) Ss 
+(Mi—Fs/2)/s —(Pe—- VAS 8V 4; 
4B4a=F+ Rn /2—-(hths/aVs 
“hat Fs/2)/a + (ht Aivavsvu,) 
where /, Ff, , etc., are rational. Let ~' (4 4s) be the sum of the 
four expressions in (70). Then, because these expressions are the 
roots of a biquadratic, ¥ (44 43) or 47h + 47%, /s 381, must be 
rational. Suppose if possible that ,/s; cannot have its value expressed 
in terms of ,/sand fz. Then, because fs / 81 is not rational, 
F;= 0. By (68), this implies that » = 0. Let 
| 
f (71) 
(45 4;2?=L+h/z2+ (n+ hLVavVvs 
+t(it+Ls/4/a%uat lt Uva s/s, 
