414 PRACTICAL MATHEMATICS 



Now h is a given increment in the value of x, and in conse- 

 quence cannot be taken as becoming infinitely small. We have 



therefore to find the best value of h -^ or &A for a definite 



ox 



value of h. 



Now u x+h = f(x + h) 



-/(a) + */ + / -!- /* + 



and (1 -H A)i, - (1H- *A + -r- + -r- 4 



or 1 + A = e* A 

 Hence &A = log e (1 -f A) 



or /?. J? = AM X - ^AX + ^A 3 w z - j 



where Aw,,., A^, A 3 Ma., A 4 ^ represent the successive differences 

 running downwards in a diagonal direction from the term u x . 



As u x represents any term, the value of h-^- can be obtained for 



doc 



any value of x given in the table, provided that the successive 

 differences corresponding to that value of x are accessible. 

 If u x _ b is the value of y preceding u x , 



Then u sa _ h =f(x-h) 



and also u x _ h = u x $u x = (1 $)u x 



h 2 h s 



Now /(* - h) = f(x] - hf'(x] + ^f"(x) - .JL. /" H- . . . 



/r>A 2 h 3 A 3 



and l- 



or l- = e- A 



Hence - AA = log e (l - 8) 



S 2 8 3 8 4 

 6 T "3" T 



