The equations of all analytic minimum curves may be written in terms of a complex parameter s and an arbitrary analytic function F(s). The curve will be algebraic when and only when F(s) is an algebraic function, and then the rank, order and class of the curve can be easily expressed as the orders of three functions, #(s), 4(s) and X(s), (see ? 1), of the general form f [s, F(s), F'(s), F"(s)]. It is the purpose of this paper to give the developments in series of