RESOLUTION OF ALGEBRAICAL EQUATIONS. 16E 
for T, its value as furnished by (7); the expression on the right hand 
side of (10) being clear of both the surds T and T,, except as these. 
surds are implicitly involved in V. Hence the surds T and T, are 
similarly involved in f (p): which is contrary to hypothesis. There- 
fore the equation (3) cannot subsist. 
PropositTion XIV. 
Let the equation, X=, be an algebraical equation of the fifth 
degree, in which the coefficients of the powers of & are rational func- 
tions of a variable p ; X being incapable of being broken into rational 
factors, that is, factors having the coefficients of the powers of & ra- 
tional. Then, should the roots of the equation, X —0, admit of being 
represented in algebraical functions, they are all contained in the ex- 
pression, 
1] -_ 
f (p)—A+(A, +B, /C) (D+D, VC) +(Ay+B, VC)(D+D, VO) 
HAR. 70D = D0) (4B. /C) (DD, VC}; ) 
where C, D, D,, A, A,, B,, A,, B,, &c., are rational functions 
of p. 
For let f(p), a root of the given equation be reduced (Prop. XII.) 
to a simple integral form, containing no surd such as Y in equation 
(1), Prop. XII., nor any surd, which, while not subordinate to any 
surd in the function, is of the form shown in (2), Prop. XII., and 
having no two surds similarly involved in it. Take Y, a surd in 
J (p), not subordinate to any other in the function. Then, if we 
consider the manner in which the terms of the series, 
BGP lk ele), Bey. aus PEM a Wore igre Na) 
in Prop. VIII, are formed, it appears, that, in an equation of the ¢™ 
degree, the reciprocal of the index of Y is a measure of ¢. Hence, 
in the case before us, the index of Y is 1; and from this it follows 
that the series (2) is reduced to the two terms, 
a—f(p), X. 
Besides Y, there can be no sard in f (p), which is not a subordinate 
of Y; for, if U were a suri in f(p), distinct from Y, and not sub- 
ordinate to any surd in / (p), then, since the coefficients of the dif- 
ferent powers of « in X are rational, the surd U disappears from X ; 
consequently (Prop. IX.) the index of U is the same with that of Y, 
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