CHAPTER VIII. 



SUCCESSIVE DIFFEKENTIATION. DIFFERENTIATION OF A 

 FUNCTION OF TWO VARIABLES. 



124. WE have, in Art. 77, defined the second differential 

 coefficient of a function to be the differential coefficient of the 

 differential coefficient of that function. The differential 

 coefficient of the second differential coefficient has been called 

 the third differential coefficient, and so on. We are now 

 about to give another view of these successive differential 

 coefficients. 



125. Let y=f(x), 



therefore &y=f(x + ?t)f (#) . 



In the right-hand member of the last equation change x into 

 x + h and subtract the original value ; we thus obtain 



-f(x + K)- [f(x + h) - 

 or f(x + 2h)-2f(x + K) +f(x). 



This result, agreeably to our previous notation, may be 

 denoted by A^Ay), which we abbreviate into A 2 ^. Hence 



A'y =/(* + 2h) -2f(x + 70 +/ (x}. 

 Similarly A (A 2 ^) or A 3 ?/ will be equal to 



- 2f(x + 2h) +f (x + K) 



that is, A s # =f(x + 3) - 3/(a; + 2&) -f 3f(x + A) - 



