RESOLUTION OF ALGEBRAICAL EQUATIONS. 163 
When U is changed into the negative expression, —U, let the terms 
in (3) become, 
x SOMME SS ORE SG a MU AWAY 5) / (3) 
and, since F,(2) must be equal to one of the terms in (8), assume ‘on 
the principle pointed out in Prop. XI.) that F, (%) = *X. Take a? 
a power of x in F, (x), such that some power of Y is present in its 
coefficient E; and, E, and E, being the corresponding coefficients of 
az in ,X, and X, let E, E,, and E,, satisfying the conditions of 
Def. 8, be, 
Ho Be Boy ee eae, 
E,=6 +6.Y, + 4,Y, + &., 
E, =f + Be (aye) thon (eve) + &e. ; 
where B,, B,, &c., none of them zero, are clear of the surd Y; B 
also being clear of Y ; and no two terms in the series, Y° , Yon eel 
are identical with one another; Y,, 6, 6,, &c., being what Y, B, B,, 
&c., become in passing from F,(x) to ,X,; and ,Y, B, B., &e., 
what these same quantities become in passing from F, (a) to x 
Then, because ,X, and 'X are each equal to F, (7), they are equal to 
one another. This implies that E, and E, are equal to one another ; 
but (Prop. XIII.) bY, is not equal to B, ( LY). It may be shown, 
however, exactly as in Prop. II., that the terms, 6,.Y,, 6, Y,, &c., 
taken int some order, are equal to the terms, (, (y ) pe (axe Ny &e., 
each to each; and, if the steps of the demonstration be referred to, it 
will be seen, that, since equation (:) subsists, b, Y, must be equal to 
to the term /, Cx) : which is impossible. Therefore U cannot be 
a chief subordinate of Y ; and T is the only chief subordinate of Y. 
Again, suppose that U is a chief subordinate of T, with the index a5 
h 
and, when U is changed into z,U, z, being a p™ root of unity, dis- 
tinct from unity, let the terms in (3) become, 
ee USE BES OS NA TAS RG) 
the surds Y and T at the same time becoming y and?¢. Thenif E, 
be the coefficient of x™ in , X, we may put 
