40-2 Mr Murphy on the Inverse Method of 



Recurring now to Equation (2), let f,S be represented by F(-.r) 

 so that F' denoting the derived function of F, we have S=—F"( — x); 



1 P 



.-. tf(t)= -coefficient of - in F'(-.v) .h.\. ^ 



= - coefficient of - in F'(-x) . (h. 1. — - h. I. ^1 , 



and, by a property of equations demonstrated in my former paper in 

 this Volume (p. 138.), this gives us 



f{f)=l{F(a 1 )+F(a l ) + ...+F{a,)-F(b i )-F{b i ) -F(b„)\. 



If we substitute differential coefficients for integrals, a process similar to 



that by which S was found, will determine S'; the value is given by the 



(IT 

 equation S't* = — fit" —r-, the integral being corrected by the condition, 



that S'=R-A when t—1, and we may verify the results by adding the 

 values of S and <S" which will give S+ S' = R .t". 



(2.) On tlie separation of the positive from the negative powers, 

 of the variable ; in functions generally. 



In the preceding example, the form of the function Rt" contributed 

 materially to facilitate the calculation of the parts which involved the 

 positive and negative powers of x respectively. As on many occasions 

 it would be useful to effect this object, we may here insert a general 

 method founded on very simple principles. 



First, let us seek the term independent of //, in a function of // 

 containing both positive and negative powers of h, but composed of a 

 finite number of terms. 



Represent by F(h) the given function, and let n be a number greater 

 than any exponent of h, positive or negative which it contains. 



Let the « lh roots of unity be 



1, a„ a 2 a„_, ot 1, e» , e » e » 



