THEORY OK ERRORS. IxXXV 



b. Probable, mean, average, and maximum actual errors of interpo- 

 lated logarithms, trigonometric functions, etc. 



When values of logarithms, etc., are interpolated from numerical tables by means 

 of first differences, as explained above, the probable and other errors depend on 

 the magnitude of the interpolating factor. Thus, the interpolated value is 



7! = 7'i -f- (r'2 — 7'i) ^ 



where r, and 7',. are consecutive tabular values and / is the interpolating factor. 



For the species of interpolated value wherein the quantity (7-2 — z'l) t is not 

 rounded to the nearest unit of the last tabular place (or wherein the next figure 

 beyond that place is retained) the maximum possible actual error is 0.5 of a unit 

 of the last tabular place, and formulas (i) and (3) show that the probable, mean, 

 and average errors are given by the following expressions : — 



e^ = ^ (i — /) for /between o and \, 



=z i — i s/zt (i — /) for / between \ and §, 

 =z\ t for / between f and i. 



S I - (t - 2 tf 

 \ 96(1-^)^ 



I _ (l - 2( f 



24 (i - ^ 



\3 



for / between o and \, 



— ^ — — T-^ for / between \ and i. 



24 {j.—t)t 



It thus appears that the probable error of an interpolated value of the species 

 under consideration decreases from 0.25 to 0.15 of a unit of the last tabular place 

 as t increases from o to 0.5. Hence such interpolated values are more precise 

 than tabular values. 



For the species of interpolated values ordinarily used, wherein {lu — v^ t is 

 rounded to the nearest unit of the last tabular place, the probable, mean, and 

 average errors are greater than the corresponding errors for tabular values. The 

 laws of error for this ordinary species of interpolated value are similar to but in 

 o-eneral more complex than those defined by equations (i). It must suffice here 

 to give the practical results which flow from these laws for special values of the 

 interpolating factor /.* The following table gives the probable, mean, average, 

 and maximum actual error of such interpolated values for /= i, i, ^, . . . tV- ^^ 

 will be observed that t =: i corresponds to a tabular value. 



* For the theory of the errors of this species of interpolated values see Annals of Mathematics, 

 vol. ii. pp. 54-59. 



