This is a reprint of a 1976 paper that appears in an inaccessible Brazilian journal and has become very looked after. It deals with the problem of determining a convex plane domain from the existence of infinitely many over determined Neumann eigenvalues. Recent related work in magneto hydrodynamics of Vogelius and other applications are closely related to this result. The more general result appears in J. Analyse Math 1980 and Crelle l987. See Zalcmain's bibliographic survey of pompeiu problem for other references.