Resolution of Algebraical Equations. 151 



SECTION VI. 



We shall here add a few theorems, connected with the present 

 subject. 



To transform the definite integral of f (x), taken between two 

 limits, which are the two roots of given equations; to another, in 

 which the limits are known quantities. 



Suppose <p(x) = o, F(x) = are the two equations, the roots of 

 which, are to be the limits of the f f{x) ; and first, let us consider 

 the value of this integral, taken from x = 0, to x = the root of the 

 equation (f>(x) = 0; this value is a certain function of that root; 



and is therefore, by Section (2), = coefficient of - , in - 1. 1-^ 



x x 



multiplied by the derived function, namely, byf(x); and apply- 

 ing the theorems given in Section 5, it follows that, ff(x) is 



= _ /• 6 9 ./(6 9 ).l.{0(e fl ).6-'' } 



2 71-^/"^! 

 the limits being and Stt^-I. 



Similarly, ff(x) from .r = to x --= root of F{x) = 



H 

 -1 



between same limits. 



= _ f e B f(e°)A.{F(e«).e -« i 



Hence, if we subtract, we get ff(x) between the two pro- 

 posed limits, viz. 



f e'./ie") , F(e e ) 50 = 



the forms of the functions being such as those already noticed. 



