cxc Tables for Statisticians and Biometricians [XXXVIII XLI 



Hence, if ai, ctz, 03, a 4 and a 5 be the abruptness-coefficients for absolute values of 

 the five initial frequencies, they will be known in terras of HI, n 2 , n 3 , n^ and r? 5 , 

 being given by linear relations. These are provided in Table XL (pp. 227 229), but 

 they are, as we shall see, only auxiliary tables, and the reader need pay no attention 

 to them unless for some special purpose he desires to find the actual values of the 

 abruptness-coefficients at x = 1. We can now put our formulae into the following 

 form in working units, i.e. the subrange h as unit, where *>/" is the crude moment- 

 coefficient of the frequency N % about a; = 1 the start of this frequency 





4- 



in i 



"2 ~ J 1 -! 



_t_ _ . . 



720 + 30240 



120 3024 



lit /AT" m \( '" 1 '" 



V 3 



/// /AT \ / "' 1 "' i 7 



4 = (J\ ??i) I Z>4 2 /'2 + yjfl 



40 504 9600 



+ 



1440 



...(iii). 



Here fig" is the corrected sth moment-coefficient of the whole frequency N, about 

 # = 1 , or the non-asymptotic terminal of the first subrange. But we note two points : 



(a) The first series of terms for each moment-coefficient are the moments of 

 the remainder, Nn^, of the frequency with the usual Sheppard's corrections. 



(6) The rest of the formulae consist of two parts, (i) the moment of the first 

 asymptotic subrange, and (ii) the abruptness-coefficients for the start of the second 

 subrange. Both these parts are linear functions of the frequency of the first five 

 subranges n\, tiz, ^3, ^4 and ?? 5 . Call their contributions to N/j. g "', K g , then our 

 formulae may be rewritten in the following manner : 



We shall have 



tf" = (N- 



.(iv)*. 



where the values of the constants q e g f , (,e s ", q e g '", q e s iv , q e g v are provided in Table XLI 

 (pp. 230231). 



The process of determining the asymptotic abruptness corrections is now fairly 

 simple : 



(i) The computer omits the first subrange frequency and computes the 

 moments of the remainder N KI about the end of the first subrange. 



* Should the reader desire to work in absolute values and not the subrange h as working unit, he 

 must use the formulae 



tyt/" = (N- j) V" + hK, , tf /I,"' = (N- ,) ' - 



