METHODS OF INTERPOLATION. 339 



US take the fourth differences of the terms in column (1), by means of 

 the formula — 



J4 = 6 Us — 4 {U2 + th) + {Ui + ^l5) 

 The results are entered in column (6). But equation (84) shows tliatthe 

 probable error is approximately equal to the probable fourth difference 

 multiplied by 0.11952. Hence, if we multiply eacli of the values of ^4 

 by 0.11952, we shall get a system of errors which may be taken to rep- 

 resent the actual errors of the given series (1). But as their sequence 

 will be very irregular, we must take the average, without regard to 

 sign, of a group of adjacent ones numerous enough to get a fair average 

 value, and this will represent the " mean of the errors." We know from 

 theory that the "mean error" is equal to the "mean of the errors" mul- 

 tiplied by 1.2533. Hence, we can get the mean error by taking an av- 

 erage value of J4 without regard to sign, and multiplying it by — 



0.11952 X 1.2533 = 0.14979 = 0.15 nearly. 

 If the average of fifteen values of /J4 is used, the process amounts sim- 

 ply to taking the sum of every fifteen adjacent terms in column (6) with- 

 out regard to sign, and dividing it by 100. The results are given in 

 column (7), for the ages 27 to 82. For instance, at the age 40 the sum of 

 the nearest fifteen terms in column (6) is 3.95, and dividing by 100 we 

 have 0.39 as entered in column (7). To get the first seven and last seven 

 terms of this series, an average of less than 15 values of J4 was employed.* 

 . The mean errors thus obtained are denoted by £2, to distinguish them 

 from the theoretical mean errors ei. The differences between the values 

 of £1 and £2, though considerable, are sometimes in excess and sometimes 

 in defect, and are, perhaps, no greater than we ought to anticipate, from 

 the usual discrepancies between theoretical and actual systems of errors. 

 It is thought that this method of obtaining the mean error may be use- 

 ful, at least for purposes of rough estimation, in those cases where the 

 more exact method cannot be applied. 



To find the probable errors directly from the average values of J4, we 

 should multiply tne latter by — 



0.14979 X 0.67449 = 0.10103 

 that is, the probable error is about one-tenth of the average fourth 

 difference. 



INTRINSIC WEIGHTS OF OBSERVED TERMS. 



We have hitherto proceeded on the assumption that the terras in- 

 cluded by an adjustment- formula are all equally liable to error, or devi- 

 ation from the true law of the series. It is possible, however, to assign 

 to each term its own proper liability to error; that is to say, its own 



* It would have been better to have obtained all the averages from only nine or 

 eleven values of A4, instead of fifteen. Owing to the general curvature of the series, 

 the average of a group of terms differs a little from the normal average of the middle 

 term, and this error increases nearly as the square of the number of terms in the group. 



