Resolution of Algebraical Equations. 133 



verging form when we make use of the ascending powers of the 

 least root, the given series arithmetically designates the least root. 

 And thus it is that Lagrange's series which, as it will be hereafter 

 seen, coincides with that obtained by the present method, repre.- 

 sents arithmetically the least root of the proposed equation.* 



SECTION II. 

 To find any function f f(x) of the root of a given equation 



(x) = 0. 



From f (0) subtract the coefficient of - in f ' (x) . I. 5i_J ; 



f (x) being the derived function or differential coefficient of f (x), 

 the remainder will be the required function. 



After the examples given in the former Section, many will 

 not be necessary, in illustration of the present principle; we 

 shall take but one, viz. to find the value of x n when x = c.e', 

 n being positive. 



The required value of x", by the above principle, is the coefii- 



1 c 



cient of - in — nx"- 1 }. (1— ~.e T ); and expanding the log. and ol»- 

 x zc 



When the first power of .r does not enter the given equation, put x = a -\- z, so as to 

 introduce a first power of z; and then, let z be treated as the unknown quantity. 



* f( x ) i s supposed to contain only positive and integer powers of J; should it be other- 

 wise, we have only to put x= a + z; the same is to be observed of <p (x). 



