140 RESOLUTION OF ALGEBRAICAL EQUATIONS. 
be neither zero nor unity, | write V, for Ur: ; and, as above, all the 
powers of U occurring in F, (z)can be made to disappear; powers of 
V; being introduced into the function in their room, but no powers of 
any other surd that was not previously in the function. Let the 
function become, in consequence of this change, F’. (x); and let the 
coefficients of the several powers of % in Be (x) and F’, (x) be sup- 
posed to satisfy the conditions of Def. 8, as was the case with F, («). 
Then, since V and Vj are respectively powers of surds that were 
present in F,(”), but do not remain (except as implicitly involved in 
V-and Vi) in F.(w); and since F, (a) is (by hypothesis) in a simple 
form, F, (w) and F. (“) are also in a simple form. In changing 
F’. (x) into F’, (x), we assumed that 7 was not zero. This may now 
be shown to be case. Equate the coefficients of 2 in F’.(«) and 
“£(®); this latter expression being what fe («) becomes when Y is 
eliminated, and powers of V introduced in its room. Then, 
] 1 k (a-1) l 
Hs ae +6 VU +... =.....+6 VU 2 a + &e. 
we deb) Ve aD ay ee 
k(a-1) 1 
Tperefarcuby or Ieee oer ek a 
If 7 were zero, this would make 2“) equal to unity: which, 
since the numbers, 4, a—l, are less than s, and 2 is an s‘* root of 
unity, distinct from unity, is impossible. Therefore / is not zero ; 
and hence E, (x) can be exhibited in the form E. (x). Equate the 
coefficients of «* in E. (x) and f, (x); this latter expression being 
what “fe (~) becomes when Y and U are eliminated, and V and Vj; are 
introduced in their room. Then the equation (6) still holds. But 
such an equation implies, that, z being an s‘ root of unity, and 
2 being a o* root of pes distinct from unity, c=s. a sup- 
Dees Vv ae occur in its oP paws in any term of E, (x), so that 
bg ne va is a term in the atigect of some power of x. Then, 
by reasoning as above, we get 
kh(a-1) lb 
l—z itty Z1 Lh Ou WR EE SE, Ae (7) 
