RESOLUTION OF ALGEBRAICAL EQUATIONS. 137 
nr factors of Fo (x)-being x—?, , ®—$2,...... ,&— |, constitute the 
nr 
‘Series of the unequal cognate functions of f(p), obtained by assigning 
definite values to all the surds which are found in F2 (2), [these being 
surds of necessity present in f (~)], and taking the cognate functions 
without reference to the surd character of these surds. In the same 
manner in which F, (x) was derived from f (p), and then F, (a) from 
F, (a), derive F; (x) from Fy, (x), and Fy (x) from Fs (x), and so on, 
till an expression F, (x) is reached, in which the coefficients of the 
powers of x are rational. The expression F, (x) shall be identical 
with X. : 
For, if the factors of F, (x) be, x—%1, ©— o2,......; a 
then, since the coefficients of the powers in # in F, (x) are rational, the 
functions ¢) , ?2,... ..... 5 Pun constitute (Prop. VII.) the entire series 
of the unequal cognate functions of f(p). But the entire series of 
the unequal cognate functions of f (p) is identical (Cor. 6. Prop. 
VI.) with the series of the roots of the equation, X=0. Therefore 
F,, () and X are identical. 
Proposition IX. 
In the series, in Prop. VIIL., 
2—f(p), Fy (x), Fe (x), .. ... by Wea (MOT NG G83 ace (1) 
let the factors by whose continued product F.+(w) is generated, be, 
2 3 Ss 
LD AMED: Boel OK Ge) Nigets PSN ig Re cnt al D8 GF) 8 tec enter (2) 
2 3 ‘ i 
where F. (wv), F.(«), &c., are what F, (x) becomes, on substituting 
successively for Y, a surd in F(z), not subordinate to any other surd 
in the function, and having the index <, the values 2 Y, oii S&C. ; 
2 being an sth root of unity, distinct from unity. Let U be a surd in 
F, (a), distinct from Y, and not subordinate to any other surd in 
F.(x), with the index = Then if the surd U disappear from 
F (a), ois equal to s, and the surds Y and U are (as we may ex- 
aril 
press it) similarly involved in the function F, (x): by which we mean, 
that, when the function is arranged according to Def. 8, whenever one 
of them appears in the function in any of its powers, it occurs mul- 
