332 METHODS OF INTERPOLATION. 



from the values wbich would be obtained if we neglected the principle 

 of continuity in the adjusted scries. In that case, denoting the proba- 

 ble error of an adjusted term by £0? we should have — 



'f = V/o^-f2(ii^+//+/3'+ -\-lJ) 



and the system of weights which renders this ratio a minimum is that 

 of the least-square formulas already referred to. 



The first two and last two terms of a given series are not reached by 

 any of the formulas of Table III, and it is impossible to adjust them 

 with the same accuracy as other terms. The best we can do, perhaps, 

 is to apply formulas (60) and (01), or sometimes to use only four given 

 terms instead of five, in which case we get by the same method-^ 



Ui= 2^ (19 Wi + 3 % — 3 u-j + Ui) (93) 



t^,2=^(3wi+llM2+9«3-3w4) . (94) 



When the law of the given series varies so rapidly that any five con- 

 secutive terms cannot be regarded as approximating closely to the form 

 of a curve of the third degree, we may liave recourse to adjustment- 

 formulas of the* nature of (22) and (59), which assume that the series is 

 of an order not higher than the fifth. Then, by processes precisely 

 analogous to those by which formulas (84) and (69) were demonstrated, 

 it can be shown that the probable error of a term bears a fixed ratio to 

 the j)robable value of the corresponding sixth difference, namely, 



£ = . 032898 (Je) 

 that the best adjustment-formula is consequently the one which renders 

 the probable sixth difference of the adjusted series a minimum j and that 

 to include only seven terms, this formula is — 



u,= g^-^ I 1959M4 + 825 {Us + %)- 330 [u^ -f tic) + 55 («i + u,) | (95) 



or, to three places of decimals, 



Ui = .040 ?f4 + .270 («3 + ^^5) — .108 {V2 + Kg) + -018 {iii + u^) 

 The weights of formula (95), taken together with the twelve nearest zero- 

 weights, constitute a series of the sixteenth order. It has been found 

 that the ratio of the probable errors of the adjusted and unadjusted 

 terms is — 



-'= 0.232 



e 



In the case of formula (70), for adjusting a double series, the weights 

 of the formula, taken together with the forty nearest zero- weights, con- 

 stitute a double series or rectangular table of forty-nine terms, whose 

 complete fifth differences zJj+s are equal to zero. The employment of 

 such an adjustnient-formula as this involves the assumption that the 

 true law of the given double series is such that any rectangular group 

 of nine terms will satisfy the condition — 



4+2 = 4 U5— 2 («2+ «4+ Ue+ Us) + («i+ U3+ M7+ Wg) = 



