Singular points of ordinary linear differential equations

in which the functions X;ff(Z) are anaZytic or have a pote at Z oo, will transform (1) into a system of equations (1) with coefficients aij(z) of the same form (2), although q is not necessarily equal to q. Any such system of equations (1) will be said to be equivatent to (1 ) at z oo. The first part of the paper deals with the determination of the simplest system of differential equations which is equivalent to (1) at Z oo. In §1 an important Lemma on AnaZytic Fqbnctions is stated and proved. The applica-