362 Mr Murphy on the Inverse Method of 



of in Part II, and it has been stated in the Introduction, to 

 what classes of natural phenomena, the respective parts corres- 

 pond. It is obvious that the form of <p{x) when rational is 



— + ^ + - % +&c. in Part I. We shall now proceed to the prin- 



x x or 



ciples, by which we may revert from the given function <j>(x) to 

 the unknown function f{t). 



(2) Inverse Method, for Rational Functions. 



5. When the known function <p(x) is rational, seek the co- 

 efficient of- in 0(x).f x ; dividing it by t, the quotient will be 

 the required function f(t). 



To prove this, suppose a to be any negative number ; the co- 

 efficient of - in (j>{x).t-% which we shall represent by T is also 

 the coefficient of- in <p(a)J-°. We shall get by actual integration 

 from t=0 to t=l, the equation, 



f^-'-'-t^ <*>- 



and taking the coefficients of - at both sides 



f t T.t" l = coefficient of i in J^ (2) 



ABC 



Now, by supposition, <f>(x) is of the form — + — t + ^ +&c. 



Consequently, 



x-a [a a* a 3 J [x x of J 



