RESOLUTION OF ALGEBRAICAL EQUATIONS. 143 
is a subordinate of Y) is impossible. Hence at least one of the terms, 
b, , b3, &c., as b,, does not vanish. But this leads to the conclusion 
that all the terms, 
Geity sala ia rate) 5 /Oiy 4) 3.< es ols coke noeiets (6) 
except b,, must vanish. For, from (3) and (4), we have, 
Yr 1 
Wee ee N Pe TL + Ge. 2G: 
: 1 ‘ 
From this equation eliminate the surd QS, in the same way in which 
X, was eliminated from equation (4) Prop. I. The result is, 
OEY EO He bY C Es Se. 0 
The conditions necessary in order that E and E, may both vanish, are, 
Cc i 
bb YT =kQ’, 
i 
RYT = 7'O™ 
k and q being constant quantities ; and these equations give us, 
Henil == bal: 
which, T° and T being distinct powers of T, not exceeding the 
(c—1)*h, is (Cor. 1, Def. 9) impossible. Therefore 6, is the only term 
in (6) which does not vanish ; and, from (8), (4), and (5), 
s cs s es+a s es+m 
ee Os Bibel ist bse + &. = Q...... (7) 
If fm be a term in the series, h, h;, &c., which is not zero, all the 
other terms in that series vanish. For, if 4, be another term, let 
es+e=weor+B, 
and,cs+m=wi,o+ 6; 
where £ and 6 are whole numbers, less than c. Then, since e and »» 
are not equal, and each of them is less than oc, # and 3 are not equal, 
And so likewise as regards the other terms. Therefore (Cor.1, Def. 9), 
all the coefficients, b, 4., 6° hm, &c., in (7), must vanish, except 
the one occurring in the term which is equal to Q. But hm does not 
vanish. Therefore all the terms, 2, h,, &c., except hy, or (as we 
may write it) H, must vanish ; and Y is reduced to the form, 
1 
me i 
