OF THE HIGHER DEGREES, WITH APPLICATIONS. 119 
} 2 2 4 
t As = Me e a2 Ae = hy Al, 
Sy ene eae 
2 
Be, 4g = 41, the 4g = & Aj, t (63) 
Deez db, 2 5 4 
therefore 42 = a 41, a2 42 = hy 4, 
5 3 5.4 5 3 
i) de = 4; ; ho ds = @ dy A J 
5 2 : : ; 
Now a; 4; , being equal to 42, is a root of the equation ¢ (x) = 0. 
10 6 FANS ea A é i re : 
And a 4;, involving only surds that occur in 7, is in a simple 
Bi Sete, 
state. Therefore, by Prop. LII., az 42 is a root of the equation 
. 5 4 5 4 ai) 8 
¢ («) = 0. Therefore f, 4), and therefore also hz Jz or e; 4; , are 
roots of that equation. Hence all the terms 
Ne eM ae ees 
Ay » A 4 » #1 Ay ’ hy Ay 5) (64) 
are roots of the equation ¢ (x) = 0. But a, e, Ay, are all 
distinct from zero; for, by (63), if one of them was zero, all would be 
1 
zero, and therefore A? would be zero ; which by §6, is impossible. 
From this it follows that no two terms in (64) are equal to one 
; 5 42 Ae a 
another ; for taking aj 4; and e; 4), if these were equal, we should 
’ / 
have q¢ 47 = a, ¢ being a fifth root of unity ; which ; which by 
§8, is impossible. This gives the equation ¢; (w) = 0 four unequal 
roots ; which, beeause it is of the second degree, is impossible. 
Therefore the first term in (55) is not equal to the second in (39), 
In the same way it can be shown that it is not equal to the, third. 
Therefore it must be equal to the fourth. In like manner the first in 
1 4 
(39) is equal to the fourth in (55). Because then ¢ Ae =h, 4; , and 
1 4 
ae = the Ae » hz 4g = hy 4,. But, just as it was proved in §56 
that, the roots of the sub-auxiliary ¢ («) = 0 being the c terms 
4; , 42, etc., there is no particular cognate form of #4 that is not a 
term in the series ¢, 4), ¢2 Ag, ...., @ 4, it follows that, if 
hy be a particular cognate form of H, there is no particular cognate 
form of 74 that is not equal to one of the terms hy 4; and hg do. 
Hence, since h; 4) = hy 42, HA has no particular cognate form 
different in value from hj 4;. Therefore, by Prop. III., Ai 4; is 
rational. 
9 
