404 Mr Murphy on the Inverse Method of 



it follows that when n is infinite 



A a = J 7 =/ 9 i<V) from = to G = 2t<\/^T, 



27TV — 1 



which accords with the result given in the memoir on the resolution of 

 equations, Section V. in this Volume. 



Lastly, we can now determine in any function of x as f(x), that part 

 which contains only positive powers of x, and the absolute term ; thus let 



f(x)=B +B l .x + B,x"+...+B. ] .x- 1 + B_-,.x- 2 + 6ic. 



.-. j^L t = \B + B i .l + B^+...+B_J + B-,.^...\x\l+k + /r + &c.\, 

 and selecting the parts of this product which are independent of //, we get 



/(I) 



B + B 1 x + B 2 x 2 + 8ic. = term independent of h in j 



2-*V 



7= f ^i*"'? from = ° to = 2tt\/-1. 

 / _ 1 ^9 1 — e" 



Note B. 



(1.) On the apparently improper Forms of <p{x). 



It has been shewn in the first Section, that when f(t) is any of the 

 functions usually received in analysis, <jb (x) must converge to as .r ap- 

 proaches oo , and therefore consists of essentially negative powers of x ; it 

 often happens however in the application, that (p (x) presents itself under 

 apparently a different form. Let us examine the difficulty thus offered ; 

 and the simplest mode of doing this, is to take a specific example. 



Let 0(^1+-^ ±J?- +.._+«.-' f« h.l.^i to find/(/). 



