OF THE HIGHER DEGREES, WITH APPLICATIONS. 85 
Therefore, ¢; is a root of the rational equation ¢ (#) = 0, being at 
the-same time a root of the rational (see §11) equation w (a) = 0, 
whose roots are the primitive V‘' roots of unity. Hence w (x) and 
g (x) have a common measure. But by $11, yp (x) is irreducible. 
Therefore it is a measure of ¢ (x); and the roots of the equation 
p (x) = O are roots of the equation g (x) = 0. Therefore, 
F (@, 6, etc.) = a, {¢ (t) t =) 
$13. Another corollary is, that if 
J (1, 4, ete.) = yor ie hau ty Sie.° hy SO, 
where /1, ho, etc., are clear of 1, the coefficients hy, hz, etc., are all 
equal to one another. For, by §12, because f (oi, 0, etc.) = 0, 
J (w, 01, ete.) = 0. Therefore w § f(a, (4, etc.)} = 0. In 
ol flo, A, etc.) t give w successively its nm — 1 different values. 
Then, in addition, 
mh, =h,+h,+..+hn. Similarly, nh,=h,+h,+ ..thy.*. h,=h,. 
In like manner all the terms /, ho, ete., are equal to one another. 
§14. Proposition II. If the simplified expression 71, one of the 
particular cognate forms of &, be a root of the rational equation 
F(x) = 0, all the particular cognate forms of & are roots of that 
equation. 
For, let 72 be a particular cognate form of &. By §12, the law to 
be established holds when there are no surds in 7; that are not roots 
of unity. It will be kept in view that, according to $1, when rocts 
1 
of unity are spoken of, such roots are meant as 1 ™ , m being a prime 
number. Assume the law to have been found good for all expressions 
that do not involve more than 2 — 1 distinct surds that are nct roots 
of unity ; then, making the hypothesis that 7; involves not more than 
m distinct surds that are not roots of unity, the law can be shown 
1 
still to hold ; in which case it must hold universally. For, let Aes ; 
not a root of unity, be a surd of the highest rank (see $3) in 7 . 
Then F (7;) may be taken to be the expression (1), and F' (72) to be 
the expression formed from (1) by selecting particular values of the 
surds involved under the restriction specified in §9. In passing from 
1 1 
m ; m 
71 to ra, let 4, , a, etc., become respectively 4, , ds, ete. Then 
m—1 m—2 
m{F(n)} =m 4, +44, + ete = 0, 
m—1 m—2 
and m\F (71) = he A, i. + e 4, + ete. 
