400 Mr Murphy on the Inverse Method of 



NOTES. 



Note A. — {Vid. Art. 9.) 



(1.) Application of the Inverse Method to Logarithmic Functions 



in general. 



To complete what was observed in Art. 9, let P and Q denote the 

 same quantities as in that Article, and let £ be a function of x which 

 remains finite when x is infinite, i. e. of the form 



A+- + ~ + &c; 



x x- 



p 



we are now to find f(f) when <p(a-) — R.h.\.-^. 



Now by the general theorem (Art. 5.) we have the Equation 



1 P 



tf(t) = coefficient of - in R. tr'.h. 1. ^ (1.) 



Suppose R.t" to be actually arranged according to the powers of x, 

 it will consist of two parts, one containing the positive powers of x and 

 the absolute term, all which we shall represent by S ; the other, contain- 

 ing the negative powers of x, suppose S' ; hence Equation (1) becomes 



tf(t) = coefficient of - in (S+ST) h.l. ^ 



p 



but since h. 1. ^ when expanded contains only negative powers of x like 



p 



S'. we may reject the term S' h. 1. ^ altogether, therefore we need es- 



timate the value of S only, and we shall thence determine 



1 P 



f(t) by the Equation tf(t) = coefficient of - in S h.l. ^ (2.) 



