RESOLUTION OF ALGEBRAICAL EQUATIONS. 167 
Cor. 2.—In all the cases in which the roots of a quintic equation, 
whose coefficients are rational functions of a variable p, admit of 
being represented in algebraical functions, the principles which have 
been established enable us actually to solve the equation. For, if X 
can be broken into rational factors, these factors may easily be found ; 
and thus the solution of the quintie is obtained. Should X not ad- 
mit of being broken into rational factors, assume f (p) equal to the 
expression in (1). Substitute for f(y) in X the expression to which 
# is thus assumed equal ; and let the result of the substitution be, 
nd ep ha: a 2?) 
b+(a,+6, /C) (D+D, /0)? +(a, +5, /C) (D+D, /O)* 
aay ay ae diate ee 
+(a,+6, /C) (D+D, /C)?+(a,+8,./C) (D+D, /C)*® 
where the expressions, >, a,, b,, a,, 6,, &c., as well as C, D, D,, 
are to be assumed rational. Put 
we 
@,=—0,\4,=0, 4,=—0, 0,=0, 
b= 0.0.10. — 004) 0) — 0); 
and these equations will enable us to find the unknown quantities in 
the value of f {p): it being taken for granted that the rational roots 
of an algebraical equation, having the coefiicients rational functions 
of a variable, can always be found. , 
Nore.—From what has been proved, it appears that the roots of 
an algebraical equation of a degree higher than the fourth do not, 
in the most general case, admit of being represented in finite al- 
gebraical functions ; and we have seen how an equation of the fifth 
degree, whose coefficients are rational functions of a variable p, may 
be actually solved, whenever, in consequence of particular relations 
among the coefficients, the roots are capable of being algebraically 
represented. It is easy to extend the conclusions which have been 
obtained t» equations of every degree ; and, from the principles es- 
tablished in the above Propositions, to show how an algebraical 
equation of any degree, whose coefficients are functions of a variable 
p, may be exactly solved, in all cases in which an exact solution in 
jinite algebraical functions is in the nature of things possible. This 
we propose to do in a subsequent paper. 
