184 RESOLUTION OF ALGEBRAICAL EQUATIONS, 
== 
2 Yr 
b+ by, thay, +... +GAn ; 
where 6, 61 , &e., are clear of the surd y;. The corresponding coefli- 
cient in Fy (x) is 
2 
b+ by BY) + by 2 Nn + &e. 
Therefore, b, @ 21) 4b, @ 21) y, 4 Ee. = 0. 
Since the surds present in this equation are surds occurring in 
f (p), and f (p) is in a simple form, the coefficients, 6; (2 — 1), 
by (22 —1), &c., must (Cor. 1. Def. 9) vanish separately. But, since 
2 is an 7*b root of unity, distinct from unity, r being a prime num- 
ber, none of the expressions, z — 1, 22 — 1, &c., vanish. Therefore 
6; , b2, &e., must all be zero: which is inconsistent with the assump- 
tion that the surd y; is present in the coefficient selected. Therefore 
Fj (~) is not equal to Fy (x); and we proved that it has no common 
measure with F,(«). Therefore no term in (1) is equal to a term 
in (2); and all the terms, ¢,, ¢2,....., zn, are unequal. In the 
same way it appears that all the terms, 1 , $2 ,...... , onr, are unequal. 
The terms, 1, ?2,...., nr, thus proved unequal, are the un- 
equal cognate functions of f (p), obtained by giving definite values 
to the surds in F, [which, from the manner in which F was gener- 
ated, are necessarily surds occurring in f(p)], and framing the 
cognate functions without reference to the surd character of these 
surds. For, in framing the cognate functions, $1, $9 5...... > nr, all 
the surds in F; (x), except y; , were considered as definite; and no 
numerical multipliers (such as 2, 2,, &c., in Def. 6) were affixed 
to them. If F contained all the surds in F, (x), except 4, , our 
point would be easily established. It may happen, however, that 
F does not contain all the surds in Fy («) except y,. Other surds 
may have disappeared from it, along with y,;. Let ¢ be one of these, 
if there be such: and let its index be +. ‘Then, in virtue of the s 
values that may be given to z, the cognate functions of f (p), taken 
on a non-recognition of the surd character of those surds alone which 
appear in F, must include s groups of such terms as 
1 » dp » veer oeean dur, Fed ne eee cee ret, 0+ vor COO Cee roe (3) 
In general, if'¢,%, &c., be the surds in Fy («), besides y, , which 
are not in F; and if = = &c., be the indices of the surds ¢, 4 , &e., 
1 
