136 RESOLUTION OF ALGEBRAICAL EQUATIONS. 
ed by giving definite values to the surds in F, and taking the cognate 
functions without reference to these surds. 
Prorosition VIII 
Let an equation of the ms degree, whose coefficients are rational 
functions of a variable p, be, X =0; X having no rational factors ; 
and let an algebraical root of this equation, in a simple integral 
form, arranged also so as to satisfy the conditions of Def. 8, be 
f(p) Take m, a surd in f(p), not a subordinate of any other 
surd in the function, with the index =; and let the cognate functions 
of f (p), obtained by successively changing w , wherever it occurs in 
f (p) im any of its powers, into m, 1%, Bel anette $ Zhi UW 3 
2, being an nt’ root of unity, distinct from unity, be, 
1 OLS (PD), P25 PZ y.evree ee AGHA 
Let F, («) denote the continued product of the terms, « — 9; , ©—99,, 
EARED Ts >» &—dn- The coefficients of the various powers of « in 
F, («), made to satisfy the conditions of Def. 8, are (Cor. 3, Prop. VI.) 
clear of the surd u ; and the terms, ¢,, $2,......... , ?,, constitute 
(Prop. VII.) the series of the unequal cognate functions of f (p), 
obtained by affixing definite values to all the surds in Fy («), [which 
are necessarily surds in f (p)], and taking the cognate functions without 
reference to the surd character of the surds so made definite. Should 
F, («), which is clear of the surd w, , not have the coefficients of the 
powers of « rational, let w , asurdin F, («), not a subordinate of any 
other surd in F, («), with the indev =, be successively replaced by wg , 
2 r—1 i ti 
Zp vio Wan by wee esis » 2, Uy; 2being an 7th root of unity, distinct 
from unity ; and, in consequence of these alterations, let Fy (#) be- 
2 
come successively F (a), F, (2), "E, Bsc deng ah she Ete (z); the fune- 
tions which are SAN Ae se in F, (x), becoming ¢ ci) 
P1> 92> > Pn» 1 (a), g pitti? vais? 
col ae ren °R (x), and becoming ¢ , &., 1n °F (x); and so 
Qn gn-+1 
on. Denote the continued product of the terms, F; (2), “ph (2), 0 0csaee 
Fy (x), when the result is made to satisfy the conditions of Def. 8, by 
Fs; (x), which is (Cor. 3. Prop. VI.) an expression clear of the surd 
u,, and such (Prop. VII.) that the functions, 9), $2 5...... o? [the 
