RESOLUTION OF ALGEBRAICAL EQUATIONS. “159 
1 
Ss 
d( d, 
RYT) RAS Ey, Oe ge) 
i 
‘ Leal ee a, irra) eae a 
— |T es + H,T T; ar HT Ty oe 
where C,, Co, &c., are clear of the surds V and T; and H,, H,, &e., 
are clear of the surds T and T,; the whole numbers, [dm], [dn], 
&c., representing the remainders left after the greatest multiples of ¢ 
have been rejected from d,m, dn, &e. But this form of Y is the in- 
admissible form given in (2) Prop. XII. Consequently the surd T is — 
not a subordinate of U. This leads to the conclusion that the surd T 
does not (except as implicitly involved in V) appear in the expression 
P. For suppose, if possible, that P is of the form, satisfying the con- 
ditions of Def. 8, 
N B 
re ages ae TN ee Dad Lene Oe Na egy 
where !, L,, &c., may involve the surd V, but are clear of T; and 
A B 
none of the terms L,, Le, &c., are zero; and T , T , &c., are diss 
1 
i Ss 
tinct powers of T. Then, if the form of U be, U=Q,, and if 
QQ, , when rendered integral, and made to satisfy the conditions of 
Def. 8, be written ¢, we have, by (8) and (9), 
: gl Ving 
¢=L+1L,T + 1L,T + &, 
1 
From this equation let the surd ¢° be eliminated, in the same manner 
in which X, was eliminated from equation (4) Prop. I. The result 
is, 
r B 
LK + K,L,T + K,L,T + &.—0. 
Here, since the surd T does not (except as implicitly Involved in V) 
_ appear in ¢, the expressions K, K,, &c., are clear of the surd T, 
Therefore (Cor. 1, Det. 9), the terms LK, K,, K,, &c., vanish 
Separately. But, from the manner in which K, K,, &c., originated, 
this implies that 
