THE present volume forming the second Part of my Wave Mechanics is devoted (as foreshadowed in the Preface to Part I) to the mathematical development of the general ideas underlying the new mechanics, connecting it with classical mechanics and constituting it a complete self-supporting theory. In building up the mathematical framework of this theory I have limited myself to what I consider its most essential elements, leaving aside a number of questions which have a methodological value only (such as the group theory) or which are met with in the solution of special problems.I t is my intention to consider some of these questions later on in connexion with the special problems which will be discussed in Part III (Advanced Special Theory’) ; I have carefully avoided complicating the general scheme of the theory by such special questions—with a few exceptions inserted for illustration (the relativistic theory of the hydrogen-like atom, for example).To make the general scheme more comprehensible I have not spared space, dealing with especially important general questions (such as the transformation and the perturbation theory, or the relativistic theory of the electron) at much greater length than would be necessary from the point of view of an adequate presentation to a sophisticated reader. I must cordially thank the editors for their readiness to meet my demands on space, which have resulted in a book larger than was originally contemplated. I must also thank M. L. Urquhart and Miss B. Swirles for help in correcting the English and the proofs.The present book, like Part I, is complete in itself, and can be read without acquaintance with Part I, provided the reader is familiar with some elementary account of wave mechanics, and is ready to explore its mathematical depths to obtain a profounder insight into the theory and to prepare himself for applying it to various special problems. The earlier portions of this book were written in 1931 while I was in America; it was completed in Leningrad nearly two years later. Some of the shortcomings of the book are due to this interruption and the impossibility of revising it in 1933 from the very beginning. A list of the more important references for each section is given at the end of the book; it is followed by a short index which should enable the reader to locate easily all the more important subjects treated.