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This book was first published in 1954. Its main focus, almost uniquely for its time, is the application of Hamilton's optical method to variational principles in space-time. Professor J. L. Synge here uses this method to generate a completely relativistic theory of de Broglie waves: a theory both simple and easily visualised in relation to space-time. Throughout this volume Professor Synge systematically develops this theory, engaging with readily accessible examples as well as more abstract concepts. His introduction helpfully contextualizes this study within the field of theoretical physics, whilst individual chapters focus upon the physics of rays and waves in space-time, the application of geometrical mechanics to free and changed particles, and the topic of primitive quantization. This book will be of interest to any physicists involved in the study of either de Broglie waves and Hamilton's optical method, or the work of Professor Synge himself.