Resolution of Algebraical Equations. 149 



Similarly, %■* J — l x any function f(a) of the least root is 



and 2tt y/ -lx the sum of that function of the m least roots is 



2 ?« ■■ ^^1/(0) - ff^p- .l{<p (e°) . ,-••! 



the integrals, in all these cases, being taken from 6 to + 2tt +y^T, 

 or through one circulation of x. If the functions f{x) and </> (a;) 

 contain negative or fractional powers of x, we must put x=a + %, 

 and proceed with * as the unknown quantity. 



In like manner the error made by stopping at the w th term of 

 Lagrange's series given in p. 145. 



Je X e »e _ ^ e »-i.e ]?( a + e e^ 

 the integral with respect to h commencing when h = 0. 

 With one application we shall terminate this Section. 



Let x" + a x + b = 0, to find x. 



2tt v /^1"x root =-/ e e»l.(e e + « + be- e ) 



Now / e .e e evidently vanishes between limits; 



„ /— =- r e 3e -b.e e 

 .'. 2 7T ,J — 1 x root = / -sjH j =- 



= f »_ r rt £gtl + 2 & 6(l 



2-/, e tv + ae° + b + \2 )ji 



^+J)+(»-x) 



4, 



