332 METHODS OF INTERPOLATION. 



formula, in making a distribution of population or deaths at the earliest 

 and latest ages of life, where the values vary so rapidly as to give the 

 series an exponential rather than a parabolic torm. 



But when our object is merely to graduate an irregular series whose 

 terms are all separately given, the easiest way to put it in an exponen- 

 tial form will be to take the common logarithms of all the terms, as has 

 been already suggested, and adjust them by the second and first 

 methods, and then take the numbers corresponding to the graduated 

 logarithms. The equation of the final series will be of the form 



the simplest case of which — 

 represents a geometrical progression. 



APPENDIX 11. 



Among the various methods which can be used for fixing the values 

 of the local weights in adjustment formulas, the following one is perhapa 

 deserving of especial notice : 



Assuming that the true law of a given series of numbers may be 

 regarded as algebraic and of an order not higher than the third, and 

 that the irregularities in the series are of the nature of accidental errors 

 or deviations from this true law, and that deviations of a given amount 

 are as likely to occur in one term as in another, let it be required to 

 find that system of weights which will render the probable value of the 

 fourth diflerences of the adjusted series, taken without regard to sign, a 

 minimum. 



Considering, in the first place, the most general form of an adjustment 

 formula comprising only five terms, which may be written — 



we have for the values of five consecutive terms in the adjusted series — 



u'i=zj—-^[1cu^+4:{u3+Ur,)-{n2+u^)] 



U'e=jrT^[1cUG+4:{U5-\-th)-{Ui+Ua)] 



