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To solve the problems of uniformization and moduli for Riemann surfaces, covering spaces and covering mappings must be constructed, and the parameters on which they depend must be determined. When the Riemann surface is a punctured torus this can be done quite explicitly in several ways. The covering mappings are related by an ordinary differential equation, the Lamé equation. There is a constant in this equation which is called the "accessory parameter". In this paper we study the behavior ofdoi:10.2307/1998172 fatcat:bcqb2z5cu5b35kdur7irmh43eq