VI INTERPOLATION TABLES. 



Then the formula of interpolation is 



/() = /(o) + n v. 



CASE 1. When |A"|<4 or <- - we have 



iv ^ J. ~ itj 



v = A\ /2 or ALj/2 



according as w is positive or negative, that is, according as a follows or precedes a . 

 This case occurs more often than any other, since mathematical tables, espe- 

 cially those most frequently used, generally are so extended that the second 

 differences are numerically less than four in units of the last place and thus may be 

 neglected, as the error so caused never amounts to half a unit in the last place, a 

 permissible error in interpolation. The maximum error from omission of A" occurs 

 when n = 1/2, and the more n differs from a half, either less than or greater than, the 

 larger the second differences may be and still be neglected. The interpolation factor 

 n may have any value less than unity, so far as accuracy is concerned. Still the 

 smaller the value of n the easier, usually, the interpolation, so that even in this case 

 it is advisable to interpolate from the nearest given value, thus making n less than 

 a half and either positive or negative. All the remaining cases which are mentioned 

 below, except Case 7, are for interpolation from the nearest given value. This is not only 

 advisable in order to render n smaller, but necessary on account of neglecting part 

 of the effects of higher differences, which would cause appreciable error for values 

 of n greater than a half. 



CASE 2. When |w|<s and |A ii |<4 or <-^ we have 



il 



v = A ' 1/2 + A ' 1/2 = A!,. 



& 



For all values of n numerically less than a half this formula is more accurate 

 than that under Case 1 in regard to neglecting second differences and the advantage 

 increases as n decreases. Also the smaller the value of n the larger the second 

 differences may be and still be neglected without sensible error. 



CASE 3. When |A" > -4 and |A Ui |<6 or < - we have 



rr n 



AS AS 



v = - '^ -- + n -^ = A'o + n TJ-. 



This formula is rigorous as far as second differences and should be used when- 

 ever second differences are too large for Case 2 and third differences are sufficiently 

 small to be neglected. In application, the value of A' may be formed and written 



A" 



on a line with and after AS; then, entering the table for n, take out WIT, add it 



to A'o to form v, and then, using the same table, obtain the value of nv, which is 

 added to /(a ) to obtain /(a) . 



3 12 



CASE 4. When (A 1 " | <24 or < -, and Ay | <48 or < -j we have 



fv fv 



M/2 + AL 1/2 iA\% + A!!{ /2 .AS s AS' AS 

 ~^T "6~ ~^T 2 A "~~G T' 



This formula includes the larger part of the effect of third differences. The 

 remaining part is 1/6 n 3 Ajj' and therefore decreases with the cube of n. 



3 12 



CASE 5. When |A i!i \ < 3 and |Ai, v | < 192 or < we have 



n n 



A"' 

 u . 



= A^-- + 



This formula includes the larger part of the effect of fourth differences. It 

 includes the same part of the effect of third differences as Case 4, so the remarks 

 there made apply as well to this case. 



