On Three Rational Functions on a Riemann Surface

Let y be an algebraic function of z defined by the irreducible relation

F(y, z)=A₀(z)yⁿ+A₁(z)y^n-1+...+ A_n(z) = 0                     (1)

where the A's are polynomials in z of degree, say, m. The manifold of pairs of values y, z which satisfy the given equation (1) constitute the Riemann surface of the algebraic function y, constructed in the usual manner. Any rational function of y and s is a rational function of position on this Riemann surface. The order of such a rational function is the number of places on the surface at which it takes an assigned value. In particular, y and z are rational functions of position on the surface of orders m and n respectively.