Some thought on the transverse electric modes in resonant coaxial cavities labeled TE1,0,1, TE1,0,2, TE2,0,1, TE3,0,1, etc., by Barrow and Mieher suggested several conclusions which are perhaps implicit in their paper but which deserve explicit consideration. In addition, the notation and the diagrams of the electric field configurations of these modes, as presented in that reference, cause misconceptions and confusion which subsequent papers and even textbooks are perpetuating. Actually, the transverse electric modes whose middle subscript is zero do not exist. They are limiting cases and are approached by the fields of the coaxial modes TE1,1,1, TE1,1,2, TE2,1,1, TE3,1,1, etc., respectively, as the ratio of the radii of the inner and outer conductors approaches 1. Several facts about the behavior of these modes for varying values of this ratio are presented. In particular, for a given mode, the resonant frequency of a coaxial cavity decreases as the ratio increases. In the case of a cavity of infinite length (i.e., a wave guide) the corresponding wavelengths (i.e., the critical wavelengths of the guides) approach the circumference of the cavity divided by the first subscript of the mode. Physical and mathematical arguments confirm these conclusions and make clear to what extent the Barrow and Mieher diagrams of the modes TE1,0,1, TE1,0,2, etc., are representative of actual coaxial modes. The practical importance of the transverse electric coaxial modes in ultra-high-frequency work is emphasized.