In a paper published in 1889, Hurwitz [3] proved that the holomorphic automorphic forms and the meromorphic automorphic functions associated with groups arising from the Riemann surfaces of algebraic functions satisfy algebraic differential equations of order at most three; cf. [5], [8], and [11].1 More precisely, given such a form or functionf, there is a polynomial Pf (EC [Xo0 Xl, X2, X3] which is zero for the specialization (X0, Xl, X2, X3)-(fJ', f', f1, fi). To generalize this theorem, let Z be analytically equivalent to an irreducible bounded symmetric domain in the sense of it. Cartan, and suppose that the (real) dimension of the Shilov boundary of Z is equal to dimcZ, the complex dimension of Z. Suppose further that r is a discrete group of holomorphic automorphisms of Z which is pseud-oconcave or such that Z/r has a compactification as a connected complex space whose meromorphic functions extend all of the meromorphic functions on Z/r. The main result of this paper is the following: