Resolution of Algebraical Equations. 139 



Find, by the last theorem, the sum of the values of that func- 

 tion for the m least roots, and also for the (m-1) least roots, and 

 take the difference. 



SECTION IV. 



In this Section we shall make a few applications of the present 

 method, to shew with what facility it gives various theorems of 

 Analysis. 



Lagrange's Theorem. 



Let 2 = « + hF(z) to find /(a) any function of z; put z = x + a, 

 the equation becomes x = kF(a + x); to find f{x + a), apply the 

 rule in Section (2). 



Hence • 



/(*) or /(* + a) =/(«) - coefficient of - in /'(a + r) 1. (l - hFa + x \ 



—f{a) + coefficient of - in 



]-f(a + x).F{a + x) + ±.— a fa + x. Fa + x 2 &c.i ■ 



Now if we consider f'(a + x)F (a + x), f{ a + x)F(a + x) ! , &c. as so 

 ?nany separate functions of a + x, and expand them by Taylor's 

 Theorem, it is visible that the above equation becomes 



/(*) = /(a) + hf (a) .F(a ) + A* . W {*).**)' + „ i£±^L &c . 



1.2 1.2.3 



s2 



