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JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. The Johns Hopkins University Press is collaborating with JSTOR to digitize, preserve and extend access to American Journal of Mathematics. By LOO-KENG HuA.The present paper is a revised form of another manuscript which the author had previously submitted for publication. The revision was necessary because the original manuscript contained some results (found independently by the author in sonme research begun in 1941) that have been' recently published in Prof. C. L. Siegel's paper on Symplectic Geometry.1 It is the aim of this paper to give a brief account of those results which are interfluent with Siegel's contributions. The remaining part of the author's research will be given later separately.The paper is divided into two parts: the first part (1-7) is algebraic in nature and gives a very brief descriptioni of the main theory with which the author deals. In the second part (8-10) the author proves that the spaces which play the important roles in the theory of analytic mappings have nonpositive Riemannian curvature. Thus the geometries under consideration are sufficiently regular, and the development (in broad-line) of the theory of automorphic functions presents no serious difficulties.The situation of the problem is well described by a statement due to Poincare :NOTE.-Because of the poor mail service between the U. S. anld China, a number of minor changes in this paper have beell made here, with the consenit of the editors, by

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JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. Annals of Mathematics is collaborating with JSTOR to digitize, preserve and extend access to Annals of Mathematics.

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