As part of a study of Planck’s blackbody radiation theory, H. Poincaré (in 1911–12) advanced a theory which analyzed the partition of energy between “resonators” and the kinetic motion of atoms. Resonators (the objects of Poincaré’s theory) facilitate the exchange of energy between radiation and matter, but otherwise their identity has remained unresolved. Poincaré considered resonators characterized by a particular mean energy ε/[exp(ε/kT)−1], which he showed to necessarily imply quantized energies nε (n=0,1,2,…). We additionally consider resonators characterized by a mean energy ε/[exp(ε/kT)+1], which (using Poincaré’s methodology) we show to necessarily imply quantized energies nε (n=0 and 1). Resonators are here identified with transitions between internal quantum states of atoms. This includes normal electronic atoms characterized by possible energies nε (n=0 and 1), as well as atoms populated by subatomic bosons (such as pions) and characterized by multiple occupancy of quantum states and possible energies nε (n=0,1,2,…). We distinguish between Poincaré’s theory and the closely related analysis by P. Ehrenfest of quantization amongst cavity modes. © 2001 American Association of Physics Teachers.