Equations of the form (1) sometimes arise,1: however, for which the conditions of Evans's theorem are not satisfied. Various cases in which this is true are considered in the present paper. In each instance an attempt is made not merely to prove the existence of a continuous solution, but also to determine its behavior for large values of x. The independent variables x and t will be restricted to real values. Functions of the single variable x will be considered for values of z Iying eitller on the range I xa or on the range Ib a _ b,