E. G. Brown. — Standard Functions in Interpolation. 421 



There is no doubt that Newton's method is in general 

 inferior to either of the others. This arises from two facts : 

 (1) the differences are so chosen as to depend on the values 

 following the interval, instead of on those of both sides of it, 

 and (2) the terms do not converge so rapidly as in the other 

 forms. We shall therefore not develope Newton's method 

 here, since in practical problems it is generally possible to give 

 the values on each side of the interval. 



Lagrange's and Bessel's methods employ the same differ- 

 ences — those indicated by the dotted line in the following 

 table : — 



Y_ 3 



A 1 



Y_, A 2 A 4 



A 1 A 3 A 5 



Y A 2 A 4 A 6 



/ - L o ^ u o -*o 



Interval I A x A 8 A 5 



^Y x A 2 A* 



Y 2 



Y, 



A 1 A a 



A 2 



A 1 



In Newton's method the top row of A is used, and for the 

 top interval. Bessel, however, instead of the even differences 

 takes the mean of the difference indicated and the one below 

 it — that is to say, he adds in half of the next following value 

 of odd differences. "What this comes to we shall presently 

 see ; but, meanwhile, it is evident that the two methods are 

 identical in result, so we need only take Lagrange's method. 

 Lagrange deduced the following functions as expressing a 

 result of finite differences (to the first difference add) : — 



A' 2 



- x C 1 ~ x ) JTg 



A3 



- X (1 - x) (1 + x) 



1-2-3 



A 4 

 + x (1 - x) (1 + x) (2 - x) — { 



+ x (1 - x) (1 + x) (2 - x) (2 + x) ~ 



- x (1 - x) (1 + x) (2 - x) (2 + x) (3 - x) ^ 



and so on. 



