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Excerpt from The Princeton Colloquim: Lectures on Mathematics, Delivered September 15 to 17, 1909, Before Members of the American Mathematical Society in Connection With the Summer Meeting Held at Princeton University, Princeton, N. J The formulation and first satisfactory proofs of these theorems, at least for the case where only two variables x, y are involved, seem to be ascribed with unanimity to Cauchy. For the implicit functions his proof rested upon the assumption that the function f should be expressible by means of a power series, and the solution he sought was also so expressible, a restriction which was later removed with remarkable insight by Dini. For a differential equation, on the other hand, Cauchy assumed only the continuity of the function g and its first derivative for y, and his method of proof, with the well-known alteration due to Lipschitz, retains to-day recognized advantages over those Of later writers. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.

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