330 ' METHODS OF INTERPOLATION, 



then (89) may be written — 



+ ^m+2 (Wni+24-W_(m+2)) ) 



If tlie given series Uo, Ui, Uz, «Scc., is of an order not higher than the 

 third, and if the adjustment-formula (88) is one which, as we will suppose, 

 expresses the relation between any 2m+l terms in a series of the third 

 or any lower order, then the adjusted series will be the same as the orig- 

 inal or given one, its fourth differences will be zero, and both members 

 of equation (90) will be equal to zero. But if each term of the given 

 series is liable to deviate from its normal value by an error of accidental 

 nature, whose probable amount is e, then the probable value of the 

 fourth difference of the adjusted series becomes — 



(z)'4)=eV^0^+2(Ji^+A'+J3'+ +^m42') (91) 



If now we denote by e' the probable error of a single term in the 

 adjusted series, formula (84) gives — 



e' = .11952 (zl'4) 

 Consequently we shall have — 



"^ = .11952 V^o^+2(zli2+j/4. +/i„,+2') (92) 



This is the desired expression for the ratio which the probable error 

 of a term in the adjusted series bears to that of the corresponding term 

 in the given series. Let ns proceed to compute this ratio, when, for 

 example, the eleven-term formula (73) is used, the weights being tajien 

 3S in Table III. Setting down the series — 



.135, .245, .290, .245, .135, .026, -.028, -.023, 0, 0, 0, 

 its fourth differences are found to be — 



.050, .041, -.012, -.050, -.045, .000, .004, -.023 

 and we have — 



- =.11952 V.0502+ 2(.04P+.0122-|-.0502+.0452-j-.0642+.0232) =.0187 



e 



which is the value shown in Table III. To illustrate the fact that this 

 is a minimum value, that is, less than it would be for any other eleven- 

 term formula, we will next take formula (21), in which the weights are 

 in arithmetical progression. Here the series — 



23, 34, 45, 34, 23, 12, 1, -10, 0, 0, 0, 

 has for its fourth differences — » 



44, - 22, 0, 0, 21, - 52, 41, - 10 

 and, consequently, we have — 



p/ iiQo2 



7 = ~[05^ ■^^^^' + ^ (^'■^' + ^'^' + ^^' + "^^^ + ^^'^ ^ '^^^^ 

 Again, let us try the improved formula (56). Using only three jflaces 

 of decimals, we have the weight-series — 

 .144, .238, .274, .238 .144, .038, - .029, - .028, 0, 0, 0, 



