SUCCESSIVE DIFFERENTIATION. 99 



126. By pursuing the method of the last Article we find 

 expressions for A*y, A 5 y, ... We shall not for our purpose 

 require the general expression for A"y. It will, however, be 

 easy for the reader to shew, by an inductive proof, that 



nf (x + A) +/. 



127. To shew that the limit of rr-4a i s "A 



We have, by Art. 125, 



AV =/(* + 2A) - 2f (x + h) +f(x). 

 But, by Art. 92, 



=f(x] + hf (x) + /" H + /"' 



i_. L 



6 and d l being proper fractions. Hence 



A 2 2/ = A 2 /" (x) + ~ {4/" (x + 20k] -f" (x + 0ft}. 



Divide both sides by h 2 , that is (Aa;) 2 , and then let h bo 

 diminished indefinitely. Hence we obtain 



the limit of =' 



. 



that is, the limit of . . ,, is -^ . 



(Ax) 2 dx* 



128. The result of the last Article may be generalized by 

 the inductive method of proof. Assume 



^y=h n f n (x}+h n ^(x) ................. (1), 



where ty (x) is a function of x and h, which remains finite 

 when h is made = 0. From (1) we have 



A n+1 <y = A"/" (x + h} + A n+1 f (x+K)- [h n f n (x} + A n+1 f (a?) } 

 ' = Al/ (x + A) -/" (x)} + A n+1 (^ (ar + A) -* ()}. 



H2 



