METHODS OF INTERPOLATION. 'SSI 



■whose fourth differeaces are — 



.028, 032, .005, -.022, -.070, -.014, .083, -.028 

 and consequently — 



i' = .0205 



. Lastly, let us extend the series of least-sqnare formulas at top of page 

 327 in the previous memoir, so as to include eleven terms; this gives — 



We = 429 I ^^ ^'6 + 84 (% + «-) + 69 (^(4 + -2(3) + 44 (M3 + Wg) + 9 («3 +«io) 



- 36 («i + un) ] 



and proceeding as in the case of formula (21), we get — 

 e' 11952 



7 ^ ^i2<r ^^ ('^^^ + ^^*^'' + ^^^' + ^^'^ "^ '^^^^ • 



l^ow, comparing all these four cases, and considering that the accuracy 

 of the adjustments made by any formula is measured by the smallness 



of the ratio -, we see that formula (73) is the best, (56) nearly as good, 



(21) considerably inferior, and the least-square formula the poorest of all. 

 It is not to be understood, however, that when the formulas of Table III 

 are used, the probable errors of the adjusted terms will be really as 

 small as indicated' by the ratios given in that table. In making use of 

 these or any other adjustment-formulas under the second method, we 

 are obliged to assame, first, that the true law of the given series can be 

 regarded, within the limits of the formula, as being of algebraic form 

 and of an order not higher than the third ; secondly, that tbe probability^ 

 of error, or of deviation from the true law, is the same for one given 

 term as for another; and, thirdly, that the number of terms included by 

 the adjustment-formula is large enough to make the actual distribution 

 of errors agree with that which is assigned by the theory of probabili- 

 ties. None of these three assumptions can be regarded as, practically, 

 anything move than a rough approximation to the real state of things 



e' 



m any given case. Hence, we must take the values of — not as o-iv- 



e ' ^ 



ing absolute measures of the ratio of the probable errors of adjusted 

 and unadjusted terms, but as a means of estimating the relative ac- 

 curacy attained by the use of different adjustment-formulas. They will 

 also serve to measure the relative smoothness of adjustment, since we 

 have, according to formula (84), 



l' = (^ 



that is, the probable value of the fourth difference is diminished by 

 adjustment in the same ratio as the probable error.* 



£' 



The numerical values of the ratio — given by formula (92) differ greatly 



* The actual dimiuution of the fonrtli differences, however, is found to be much more 

 rapid than that of the probable errors, and is very nearly that which theory requires. 



