with U a column vector and A and P n-square matrices. The transformation U = TU, by a unimodular matrix T is easily seen to result in an equation in U, of form (1), in which the coefficient of A is T-1AT. It is known [1] that if the elements of A and its characteristic roots are holomorphic in a closed bounded region R, then there exists a matrix T unimodular in R such that T-1AT is in triangular form. It may be remarked that the theorem is also true for the case R is unbounded.2 It is easily verfied that the method of [1] may be extended to this case without significant alteration. The only possible difficulty is the requirement [1; p. 470] of a function whose finite expansion3 is specified at a set of points which in this case may not be finite (although without a finite accumulation point). However, the existence of such a function is established by a theorem of Mittag-Leffler [2; pp. 5-6]. Another proof which may be extended to the unbounded case is that given in [3] for general principal ideal rings. It is known that the ring of all functions holomorphic in an unbounded region satisfies all the postulates of a principal ideal ring except the infinite chain condition [Cf. 4; p. 351]. However, the proof of [3] makes no use of the chain condition and so may be applied immediately. It will accordingly be assumed that the coefficients of the original equation satisfy the above conditions. In the following it will be supposed that the transformation has already been performed, so that in (1) A is triangular.