AN ELEMENTARY TREATISE ON DIFFERENTIAL EQUATIONS AND THEIR APPLICATIONS by H. T. H. PIAGGIO, M. A., D. Sc. PROFESSOR OF MATHEMATICS, UNIVERSITY COLLEGE, NOTTINGHAM SENIOR SCHOLAR OF ST. JOHNS COLLEGE, CAMBRIDGE LONDON G. BELL AND SONS, LTD, 1949 First published May 1920. Reprinted 1921, 1924, 1925, 1926 Revised and Enlarged Edition 1922 reprinted 1929, 1931, 1933, 1937, 1959, 1940, 1911, 1942, 1943, 1944, 1945, 1946, 1949. PRINTED IN GREAT BRITAIN BY ROBERT MACLKHOSE AND CO. LTD. THE UNIVERSITY PRESS, GLASGOW. PREFACE THE Theory of Differential Equations, said Sophus Lie, is the most important branch of modern mathematics. The subject may be considered to occupy a central position from which different lines of development extend in many directions. If we travel along the purely analytical path, we are soon led to discuss Infinite Series, Existence Theorems and the Theory of Functions. Another leads us to the Differential Geometry of Curves and Surfaces. Between the two lies the path fi