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This has three roots Xj(z); j = 1, 2, 3. Differential equations (1.1) can be classified into categories having solutions of quite distinct functional forms upon a basis of the configuration of these roots Xj(z) in the domain of z under consideration. To some extent this can be seen even in the elementary case of an equation (1.1) with constant coefficients. For, such an equation is completely solved by the functions eXxiz if the roots xi are all simple, whereas this set is incomplete if multiplicities among the Xi occur. A substantially complete body of theory exists for the forms of the solutions of a differential equation (1.1) in a region of z in which the roots Xj(z) are everywhere distinct [1]. Such is also the case if the z-region is one in which there are multiplicities among the roots Xj(Z), provided these multiplicities all maintain identically over the region [2; 3]. The matter stands differently for a region in which a multiplicity among the roots Xj(z) occurs, but does so only at an isolated point. It is the purpose of the present paper to give a theory for a class of differential equations (1.1) in such a region of z. The place of this theory, and its potential importance, can easily be made clear by referring to the analogous differential equation of the second order,

On the asymptotic forms of the solutions of ordinary linear differential equations of the third order in a region containing a turning point