148 RESOLUTION OF ALGEBRAICAL EQUATIONS. 
would be present in F,(v): which is impossible. Also Y, is not a 
subordinate of any surd in the equation, because all the surds in the 
equation except Y; are present in F, (x), and Y, is not in F, (x); so 
that no surd to which Y, is subordinate can appear in F, (w). Let 
then the equation, satisfying the conditions of Def. 8, be, 
AY B By 
H+H,Y a + Hy, Y Yy + &e.=0, 
where H, H, &c., are clear of the surds Y and Y,; at least one 
number in the series, 4;, 8; , &c., (say A; ), not being zero; the 
corresponding coefficient H, being at the same time distinct from 
r 1 
zero; aud no two terms in the series Y Y; , Y Yj , being identi- 
eal.. Then (Cor. 1, Prop. I.) an equation, 
ev ne ae ee Liga) 
must subsist ; where P is an expression sale f be such surds 
es occur in the expressions H, H,, &c., or are subordinates of the 
B18 
surds Y, Y;; m being either unity or zero: the term Y Yj stand- 
Aw xh deeds, 
ing as the type of any term in the series, Y Y, , Y Y_, &e,, 
after the first. But, should m be zero, equation (8) is of the form 
(3): which, since F, () 1s not equal to a term in (2), is (Cor. 1) imad- 
missible. Should m be unity, equation (8) becomes 
ie ania 
Here, by hypothesis, the numbers A—, 4;—f,, do not both vanish. 
Should the latter vanish, the equation is at variance with the suppo- 
sition that F, (a) is in a simple form. Should the former vanish, 
the equation is at variance with the fact that ;X, is in a simple 
form; which, however, 1t must (Prop. [V.) needs be. Should neither 
vanish, the equation is of the inadmissible torm (3). Hence the 
function & (p) cannot but be in a simple form. 
Cor. 3.—Should F, (#) not be equal to a term in (2), the equa- 
tions, 
F,(£) = HE Sc AM <M OS cane x Woes (9) 
F,(#) = K x Ky SGA Gis Boa RE x Kes ; 
and so on, subsist; where L is the H.C. M. of F.(#) and 1X); 
