Proc. R Soc. Lond. A 371, 67-76(1980) Recollections BY C. HERRING Department of Applied Physics, Stanford University Herring, (William)Conyers. Born New York 1914. Educated Princeton Univer. y. Research physicist Bell T'elephone Laboratories. Research on many branches of physics, particularly magnetism 1. ONE-ELECTRON ENERGY BANDS As a graduate student in the 1930s I got my principal education on the electronic uantum mechanics of solids from the Sommerfeld-Bethe article in the handbuch ler Physik. It seemed quite clear from this that the one-electron wavefunctions and nergies of a self-consistent field solution for a crystal would often have character istics intermediate between those of the tight-binding approximation and those of the almost-free-electron theory, and that therefore one would need to find new and less restrictive ways of calculating them. The Wigner-Seitz method, which had just been developed, was one such, although its initial validity was limited to states ghbouring the he=0 state of the conduction band of a non-transition metal Beyond this, one very appealing handle was to study the rigorous properties of Bloch states that could be inferred from crystal symmetry. Louis Bouckaert and Roman Smoluchowski, who were working as postdoctorals with Wigner in the academic year 1935-6, undertook to study these symmetry properties for electrons n the common eubic lattices; their work culminated in the now famous B.S.w paper of 1936. I held many discussions with them, and became intrigued by the study of the topological behaviour of the functions En()describing the energies of the bands n, and particularly by the topology of the evolution of these functions as the potential energy function, i.e. the Hamiltonian of the problem, was changed continuously. The development of this interest came while Wigner, under whom I was working as a graduate student at Princeton, was away for the second term at Wisconsin. In our long-distance communications, he encouraged my interest and made the further very fruitful suggestion that it would be of interest to supplement the study of spatial symmetry properties with a study of time-reversal symmetry, a subject to whose general formulation(as distinguished from its application to crystals)he had already made fundamental contributions. Before the end of the academic year I had picked up enough ideas to keep me busy, working at home on Staten Island over the summer. The result was my thesis On energy coincidences in the theory of Brillouin zones. This had two parts. The first was an analysis of consequences of time-reversal symmetry, and the development of criteria fo predicting when it would require the 'sticking together'of bands that would be independent if only spatial symmetry were considered. The second and longer part