(103) {^ 



STORY. — A NEW GENERAL THEORY OF ERRORS. 197 



the Cs are determined ; we may, then, take the values given by (98) 

 and (99) as approximate in all cases. 



The restriction of the development (78) oif(s) to a finite number of 

 terms is justified only by the fact that the P's that would occur in later 

 terms are all negligible, that is that 



'Pam — for a < fu, 

 \,+i = for /3 < m, 

 to the degree of approximation to which the calculations are restricted. 

 In the special case for which s = — oo and s ^= + <x> these equations 

 (103) are exactly satisfied by (89)-(91), because the suffixes of the P'a 

 in (103) are different from those of any of the P's that occur in the 

 right members of (79). 



In applying the general method to any particular set of observations 

 we have, then, to calculate, by (l)-(7) and (48) (from the observations), 

 the values of Zq, the residuals ($ = z — Zq), the /)'s, the p's, and the P's 

 with so many successive sutfixes as to satisfy ourselves that the P's with 

 suffixes greater than a certain number are all negligible ; the greatest 

 even suffix and the greatest odd suffix of the P 'hat are not negligible 

 are the values of 2 a and 2/3+1, respectively; the values of the P's 

 that are not negligible and the approximate values of the corresponding 

 Cs given by (98) and (99) being substituted in (77) determine an 

 approximate form of 6 (s) ; the positive and negative roots of 6 (s) =: 

 that have the smallest absolute values are approximate values of « and s, 

 with which more accurate values of the (7's can be calculated by (80) ; 

 substituting these values of the Cs in (77) we obtain a more exact form 

 of 6 (s), from ^ (s) = more exact values of s and s, and from (80) more 

 exact values of the Cs ; and so we continue until the variations of the 

 values of the Cs are reduf^ed to negligible quantities ; the final values 

 of the Cs being substituted in (78) give the most exact form oif(s) 

 and, by (27), the most exact form of if/ (^) that can be derived from the 

 observations. The closeness with which the conditions (81) are satisfied 

 by the final values of s and s may be regarded as a test of the goodness 

 of the observations, that is of the closeness with which the residuals 

 follow the law that they follow most closely. 



It may be observed that there is no case for which a =:: 1 ; more 

 properly, the cases for which a = 1 coincide with the cases for which 

 a = 0. 



If Pjn+i = for 1 ^ m,f(s) as given by (78) is an even function of 

 s and the relative frequencies of positive and negative residuals that have 



