METHODS OF INTERPOLATION. 301 



of the case continues, the problem of constructing'' a tabh^ of mortality 

 must be regarded as, to some extent, an indeterminate one. Not only 

 is absolute accuracy unattainable, but we cannot even decide, by the 

 method of least squares, that a certain result is the most probable of 

 any; for the true form of the function beinj^' unknown, any particular 

 residual error, or difference between the observed and computed values 

 of H term, will in general be the aggregate of two errors, one of them 

 due to the difference of form between the assumed function and the true 

 one, and the other due to the error of observation or difference between 

 the observed value and the true value. The latter portion only can be 

 of the nature of accidental errors, so as to be subject to that law of dis- 

 tribution Avhich the method of least squares assumes, and which is 

 derived from the theory of probabilities. Hence, we cannot infer that 

 because we have made the sum of the squares csf the residuals a mini- 

 mum, the resulting values of the constants which enter into the assumed 

 equation of the series must be the most probable values. To justify 

 such an inference, it would be necessary to make the sum of the s(piares 

 of the accidental i)ortions of the residuals alone a minimum ; but we 

 have no means of effecting this, for we cannot separate the accidental 

 portions from the others. When the method of least squares is applied 

 under circumstances like these, it loses its peculiar claims to theo- 

 retical accuracy, ami becomes merely a method of interpolation, whose 

 merits are to be judged, like those of other methods, by the amount of 

 labor required in obtaining the final results, and by the degree of ac- 

 curacy Avith which these results represent the observations. We may 

 in-esume that the best method of reduction for tables of mortality is 

 that which will give, in the simplest manner, a graduated series conform- 

 ing to those conditions which are known to govern such tables, and 

 representing the observations with the necessary degree of accuracy. 

 In behalf of the method here proposed, it may be said that the process 

 of computation is comi)aratively simple; that the observations are 

 represented with great accuracy throughout all the middle ages of life, 

 which is just the i)ortion where accuracy is most important in practice; 

 and that a transcendental fornuila, if it contains not more than three 

 or four constants, will be very likely to prove inferior in this respect. 



From all the foregoing considerations we conclude that a very good 

 way to graduate an experience rate of mortality for insured lives will 

 be, to forma series like (r/), expressing the probability of dying within 

 a year, at each age, and to adjust it approximately, in the first place, by 

 some formula or formulas under the second method, and then, dividing 

 the adjusted terms into the proper number of groups, to complete the 

 graduation by either the first or the third method. Treated in this way, 

 the arithmetical means of the terms in the several groups will be brought 

 nearer to their normal value than they would be if the approximate or 

 preparatory adjustment were omitted. 



In constructing a rate for general population from census returns and 



