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American Journal of Mathematics, vol. xxi. No. 2, April.—On systems of multiform functions belonging to a group of linear substitutions with uniform coefficients, by E. J. Wilczynski. In this memoir, the author attempts to prove the existence of certain general functions, studied herein, he believes, for the first time. The existence of a large and important class of these functions is demonstrated by an indirect method, which con sists essentially in generalising the hypergeometric functions in a proper manner. The work is connected in a way with the researches of Fuchs, Schwarz and Neumann (on Riemann's theory of Abelian functions, and of Klein (Math. Ann., Bd. 41). Oskar Bolza states that his principal object, in his paper on the partial differential equations for the hyperelliptic θ and σ functions, is to replace part of Wiltheiss's work (Crelle, Bd. 99, and Math. Ann., Bd., 29, 31 and 33) by simpler and more direct proofs.—E. B. Van Vleck contributes an article on certain differential equations of the second order allied to Hermite's equation. The treatment is thorough, and the work is accompanied with numerous diagrams.

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https://doi.org/10.1038/060090b0

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