158 RESOLUTION OF ALGEBRAICAL EQUATIONS. 
Therefore, from (5), Wx = wo +N, 
We Ww 
and, TROT) =H: 3 
w, being a whole number; aad H, an expression clear of the surds T 
and T,. And soon. Hence the form of (AY) is, 
cs WwW 
CAGY = Die Vier EM i Ros Siig cs oe an (6) 
neither of the surds, T, T,, appearing (except as implicitiy mvolved 
in V) on the right hand side of the equation. From the expressions 
A, and Y let the surd T, be eliminated, by substituting for it, where- 
ever it occurs in any of its powers, its value derived from the equation, 
sage) al 
eM A ye Gas ¥t locas 
and, when thus modified in form, let A, and Y, satisfying the condi- 
tions of Def. 8, become respectively P and U. Then the surd T 
cannot (otherwise than as implicitly involved in V) be a subordinate of 
U. For suppose, if possible, that it is. Put Q to represent the ex- 
pression, D+ D,V+ &c. Then 
1 
g 6 
Now, any function involving merely such surds as occur in equation 
1 
(8), exclusive of Q:. is in a simple form; for, all the surds in the 
expressions, P, U, and Q, except the surd V, are found in f (p); 
and, if an equation such as (If) Prop. I. could be formed, involving 
the surd V, that equation, when V was replaced by TT is would be 
reduced to a corresponding equation involving only surds inf (p); 
which, since f (p) is in a simple form, is impossible. Hence, since 
the surd T, a chief subordinate (on the hypothesis at present made) 
of U, is not present in Q, it is (by Cor. Prop. X.) implied in equation 
(8) that the form of U is 
iL 
Da) 
where L is an expression involving merely such surds, exclusive of 
U and T, as occur in the expressions P, U, and Q; and A is a whole 
number less than s. Restore U to the form Y; and let the surd V, 
in L, be replaced by its value in (7). Then 
