338 METHODS OF INTERPOLATION. 



Before tlie adjustment in column (3) was made, several others had 

 been made with diiferent formulas from Table III, the details of which 

 may be omitted here; but the results were that when the five, nine, and 



fifteen term formulas were used, the arithmetical means of ( - j were 



found to be considerably less than unity, namely, 

 0.40, 0.5S, 0.80 



the probable errors being about the same as in the other case; so that 



none of these adjustments satisfied the test. The mean value of ( - j 



is here seen to increase as the number of terms included by the adjust- 

 ment-formula is increased. 



The test weJiave jjroposed may be satisfied by adjusted series obtained 

 in many different ways, the problem of adjustment being really, to some 

 extent, indeterminate, as stated at page 301 of the previous memoir. 

 When the adjusted series {h) in Table II is tried in the same way, within 



the same limits of age, the arithmetical mean of f - ) is found to be 



1.05, so that that series also satisfies the test. It is believed, however, 

 that the present adjustment is the better of the two, the arrangement 

 of the weights in the formulas of Table III being such as to make the 

 closest possible approximation to the actual form of a given series. 

 And the writer would here express the opinion that, for the mere purpose* 

 of smoothing down irregularities, the second method of adjustment, as 

 perfected in the formulas of Table III, is in most cases preferable to 

 either the first or the third method, both on the ground of simplicity 

 and of accuracy. 



So long as the true analytical la®' of a series remains unknown, it 

 must be considered futile to attempt to fix a precise value for the prob- 

 able error of an adjusted term. According to the theoretical ratio in 

 Table III, the probable error of a term in series (3) is only .00185 of that 

 of the corresponding term in series (1). But the ratio is really not less 

 than about i, so far as can be judged from a comparison of several 

 different adjustments which satisfy the test proposed. This discrepancy 

 is owing to the fact already noticed, that the assumptions we are obliged 

 to make regarding the law of the series and the distribution of the errors 

 are only a rough approximation to the real state of things. 



MEAN ERRORS ESTIMATED FROM IRREGrULARITIBS OF SERIES. 



When the theoretical mean errors of the terms in a given irregular 

 series cannot be estimated as in column (2) of Table IV, either because 

 the original data are not given, or from other causes, it may still be pos- 

 sible to make a rough approximation to their amount by means of the 

 actual irregularities of the series, provided that it has not been tampered 

 with, but gives the unaltered results of observation. To illustrate, let 



