A method is presented for the analysis of data representing functions of two variables, when the response can be tabulated in a rectangular array. The procedure is based on a partitioning of the row by column interaction effects into a sum of terms, each of which is the product of a row factor by a column factor. the factors in each term are estimated by a method involving the extraction of characteristic roots. The method contains as special cases a number of procedures used for the handling of non- additivity in two way arrays. It is very useful for the fitting of empirical surfaces, but is also applicable to cases in which the data depend on qualitative rather than quantitative factors. Comparisons with other techniques are made and an illustrative example is given.