RESOLUTION OF ALGEBRAICAL EQUATIONS. 31 
1 
Qh 4/ p)°. Since therefore the coefficients of the terms, H, X,, 
Bee ia « , X,, in (6), when arranged so as to satisfy the conditions of 
Def. 8, involve only surds of lower orders than those whose powers 
constitute the factors of X,, X,, &c., and since the coefficients of 
the terms, X,,........., X%,, in (6), are all distinct from zero, it follows 
that equation (6) is of the same character ay equation (4). But 
equation (6) contains one term less than equation (4), X,, having 
been eliminated. Therefore, in the same way in which equation (6) 
was derived from (4), we may deduce from (6) another equation of 
the same character as (6), but with aterm fewer. And so on, till 
ultimately we get 
6H+/7X,=0; 
where 7 and 6, the former not zero, involve no surds of so high an 
order as those whose powers constitute the factors of X,. But 
[compare the reasoning by which equation (9) was deduced from (8)] 
this is virtually an equation of the inadmissible form (3). Hence, 
in consistency with the hypothesis that equation (3) cannot subsist, 
it cannot be supposed that the terms, X,, X,, ......... , X,, in (4), 
are all distinct from one another. Should X, then be identical with 
X,, let the two terms, BH, X,, DH, X,, be combined into the single 
term, X, (BH, + DH,). Make all other such modifications on 
equation (4) as are possible. Ultimately we get 
H+ X, (BH, + DH, + &.) + X, (CH, + &c.) + &c.=0...(10) 
where no two of the terms, X,,X,, &c., are identical. But, by 
what has been proved; this is impossible, except upon condition that 
the coefficients of X,, X,, &c., vanish separately. Put therefore 
BE pre + kel 0 2) a ody 
If we compare this equation with (2), we perceive that it is of the 
same character as (2), with this difference, that there is no surd in 
equation (11) of so high an order as some of the surds in equation 
(2). But, in the same manner in which we derived (11) from (2), 
we may deduce from (11) another equation bearing the same relation 
to (11) as (11) bears to (2). And so on, till ultimately one of the 
equations, such ag (10), at which we arrive, contains only one term 
such as X,, with no more than a single term, such as BH,, for its 
coefficient: from which it follows that B must be zero; whereas all 
the coefficients, B, C,........., H, in (2), were supposed (see Def. 9) to 
