On the Rationale of Formulæ for Graduation by Summation

Abstract

So much has been written on the algebra of formulæ for graduating an irregular series by means of successive summations, and on the theoretical errors introduced by this process, that there is possibly some danger of the underlying principles, on which all such formulæ are founded, being overlooked or insufficiently understood. In the present note an attempt is made to deal with this matter from first principles, with the object of rendering clear how such formulæ Work and why it is that they produce, with greater or less success according to the nature of the particular formula, a smooth succession of values from rough and irregular data.