Given n observations (xt, yt), each subject to an error of measurement ascertained independently, the existence of a structural relation for the true points (X, Y) is discussed. For a linear relation a Ø-number is defined from which either a confidence or a structural theory can be developed. The relation is treated in toto. Structural theory is described in terms of three hypotheses of regularity, which enable confidence and fiducial theories to be contrasted. It is shown that the theory provides a solution for the curvilinear relation wherein all the alternative hypotheses are incorporated, and it is suggested that the latter is a necessary requirement. The alternative hypothesis implicit in the Ø-number treatment of the linear relation is deduced from the general theory.