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On Three Rational Functions on a Riemann Surface
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Let y be an algebraic function of z defined by the irreducible relation
F(y, z)=A0(z)yn+A1(z)yn-1+...+ An(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.
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