INTERPOLATION. 



VII 



12 12 



CASE 6. When |Al> v |<192 or < -4 and |A V |<96 or <~3 we have 



/I* /t 



A i" A v 



-4-lf + + 



This formula is rigorous as far as third differences and includes the larger 

 part of the effects of fourth and fifth differences. Very seldom will the differences 

 be so large that the employment of this formula would produce appreciable error. 



The advantages of the above formulse over the usual forms are that the same 

 interpolation factor is used throughout, that they are more rigorous without 

 requiring the computation to be carried to extra decimal places, and that higher 

 differences may be neglected when n is small. 



For the sake of completeness there is appended the following formula which 

 includes the entire effect of all differences as high as the fifth and is rigorous for 

 all values of n and from which the preceding approximate formulse are derived. 



v = A!,- 



Aiii 

 



30 



AS 



"2 



Air 



24 



-24 



Air 



When the differences of the third or higher order are large and n is between 1/3 

 and 2/3 it is often expedient to interpolate first to halves, thus reducing both the 

 size of the differences and of n and bringing the final interpolation under a simpler 

 case. For interpolation to halves we have, as far as fifth differences, the formula 



h\ . , , A'i /2 1^ A'Q + A'i 3 AQ V + A\ T 

 f 2 "8 2 



/( 



a K = 



__ 



+ 128 



To express the final interpolation in terms of the original values of the function 

 and differences, let 



v denote the new interpolation factor, 7, the new proper variation, 



A'l/2 = 



A? 



Aiv 

 1/2 ~ 



? + 



Then our formula of interpolation becomes 



where 



v = 2n - 1 



and the proper variation, 7, is formed from the formula below, using such 

 differences as may be sensible. 



1 2 



CASE 7. When g<|n|<g, and |A Ui |<216 we have 



AU Af^ /Af/2 _ 3 A\ v /2 \ 

 2 48 " v \ 8 128 / 



and in thia case 



/(a) = /(a ) + 



AI/Z AiV 



3 A 1 / 



