406 Mr Murphy on tlie Inverse Method of 



and therefore by Art. 8. Ex. 5. 



Consider lastly the case of <p (x) = (a + x)± — (x - «)*, to find f(t). 



(Or* \ 4 

 1 1 and expanding, we have 

 W "T" 0C ! 



a 1.1 ft 2 1.1.3 ft' 



* W ~(a + *)t + 1.8" (« + *)* + 1.2.3 •(« + *)! +ttC ' 



and therefore by Art. 12. Section I. 



■ /(/) ~' ! U(h.l.*)-4 + 1.3- f t (h.\.t)i + 1.2.3- Jf(h.l.#)l KC -| 

 <"- f 2a(h.l./)- t _ (2a) 2 .(h.l./)i (2«) 3 .(h.l./)* 1 



1 f-t-" 

 = 2^4 " * (h. 1. *)* " 



(2.) Method of estimating the Values of Operative Functions. 



If any function of x be reduced to the form of a sum, composed 

 of terms which are other functions of x ; the identity thus formed will 

 continue to subsist, when x is changed into x + h in both its members. 

 Let the operation which indicates that a; is to be changed to x + h be 

 denoted by ^, let ^" denote that x must be changed to x + ?ih and 



4," , into x -\ — , then \^ n is evidently equivalent to the repetition n terms 



of the operation ^, while the latter is equivalent to the repetition » 



terms of \p' ; if beside the operation \|/' the function of x is also to be 



