Hypergeometric equations and modular equations - Euler and Gauss, Jacobi and Kummer, Riemann's approach to complex analysis, Riemann's P-function, interlude -Cauchy's theory of differential equations; Lazarus Fuchs - Fuchs's theory of linear equations, generalisation of the hypergeometric equation, conclusion, the new methods of Frobenius and other; algebraic solutions to a differential equation -Scharz, generalisations, Klein and Gordan, the solutions of Gordan and Fuchs, Jordan's solution, equations of higher order; modular equations - Fuchs and Hermite, Dedekind, Galois theory, groups and fields, the Galois theory of module equations, c.1858, Klein; some algebraic curves - algebraic curves, particularly quartics, function-theoretic geometry, Klein; automorphic functions - Lame's equation, Poincare, Klein, 1881, Klein's response, Poincare's papers of 1883 and 1884. Appendices: Riemann, Schottky, and Schwarz on conformal representation; Riemann's lectures and the Riemann-Hilbert problem; Fuchs's analysis of the nth order equation; on the history of non-Euclidean geometry; the uniformisation theorem; Picard-Vessiot theory; the hypergeometric equation in several variables - Appell and Picard. Notes on chapters and appendices. Bibliography.
