## Proceedings of the Edinburgh Mathematical Society

### 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 u1, u2, … un 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 δu1 = u2u1, δu2 = u3u2, …, δ2u1 = δu2 − δu1, 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 u1′, u2′, u3′, …, un′, whose terms differ as little as possible from the terms of the sequence u1, u2, … un, but which has regular differences. This smoothing process, leading to the formation of u1′, u2′ … un′, is called the graduation or adjustment of the observations.

(Accepted November 14 1919)