xlviii Tables for Statisticians and Biometricians [V VII 



or, to seven figures, 



r 13 (1-03467) = - -035,3847. 



We should not be correct to the fifth figure had we not used fourth differences. 



This is our "First Method," taking second and fourth differences from the table 

 itself. 



We will now use our "Second Method" or formula (v). We require to take out 

 of the table 



and 



), and Tl9 (ri), 

 and from the auxiliary Table V (p. xlvi), 



1X13, 2%is, s%i3, and 4 ^ 13 . 

 These are respectively 



-038,1455, +'029,2108, -'020,5443, +'012,5634, 

 -029,0283, +-017,9171, --008,0852, --000,2002, 

 1449,1377, -0019,9165, -0238,9979, -0007,3664. 

 Hence by (v), each by continuous process on the machine, 



S 2 r 13 (l-0) = -0041,9213, S 2 r 13 (11) = '0025,8033, 

 S 4 ri3 (1-0) = - -0004,8175, S 4 r 13 (11) = - -0001,9338. 



These values are close to those previously found directly from the table itself. 

 Substituting, we have 

 r 13 (1-03467) = -3467 (- -029,0283) + -6533 (- -038,1455) 



- -0377,4985 {1-3467 (-0025,8033) + 1'6533 (-0041,9213)} 

 + -0042,0250 (2-3467 (- -0001,9338) + 2-6533 (- -0004,8175)} 

 = - -0100,6411 



-0000,0728 



which only differs by a unit in the eighth decimal place from the value found by 

 the " First Method." 



Illustration (ii). Required r 5 (3'9746). The function lies between r 5 (3*9) and 

 T 6 (4*0), and accordingly, being on the border of the table, we cannot take out of it 

 the values T 5 (4'l) and T 6 (4'2) needful to find S 2 and 8 4 from the table; we must 

 use formula (v) or else we must use a backward difference formula like 



- 1) (0 - 2) (0 - 3) A% n + etc. . . .(viii), 

 where Az n = ,?_! z n , etc. 



