This book by the French mathematician Henri Poincaré is a seminal work in the theory of elliptic functions, a special class of mathematical functions that appear in many areas of physics and engineering, such as quantum mechanics, fluid dynamics, and electromagnetism. With elegant and concise mathematical proofs, Poincaré explores topics such as modular forms, Riemann surfaces, and hyperelliptic integrals, laying the foundation for modern research in the field. This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work is in the "public domain in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant.