On a New Method of Graduation

Suppose that as a result of observation or experience of some kind we have obtained a set of values of a variable u corresponding to equidistant values of its argument; let these be denoted by u 1 , u 2 , … u n If they have been derived from observations of some natural phenomenon, they will be affected by errors of observation; if they are statistical data derived from the examination of a comparatively small field, they will be affected by irregularities arising from the accidental peculiarities of the field; that is to say, if we examine another field and derive a set of values of u from it, the sets of values of u derived from the two fields will not in general agree with each other In any case, if we form a table of the differences δ u 1 = u 2 – u 1 , δ u 2 = u 3 – u 2 , …, δ 2 u 1 = δ u 2 − δ u 1 , etc., it will generally be found that these differences are so irregular that the difference-table cannot be used for the purposes to which a difference-table is usually put, viz., finding interpolated values of u , or differential coefficients of u with respect to its argument, or definite integrals involving u ; before we can use the difference-tables we must perform a process of “smoothing,” that is to say, we must find another sequence u 1 ′, u 2 ′, u 3 ′, …, u n ′, whose terms differ as little as possible from the terms of the sequence u 1 , u 2 , … u n , but which has regular differences. This smoothing process, leading to the formation of u 1 ′, u 2 ′ … u n ′, is called the graduation or adjustment of the observations.