A significant number of Be stars show a second Balmer discontinuity (sBD) attributed to an extended circumstellar envelope (CE). The fast rotational velocity of Be stars undoubtedly plays a significant role in the formation of the CE. However, Bn stars, which are also B-type rapidly rotating stars, do not all present clear evidence of being surrounded by circumstellar material. We aim to characterize the populations of Be and Bn stars, and discuss the appearance of the sBD as a function of the stellar parameters. We expect to find new indices characterizing the properties of CEs in Be stars and properties relating Be and Bn stars. Correlations of the aspect and intensity of the sBD and the emission in the H$alpha$ line with the stellar parameters and the $V!sin i$ are presented. Some Bn stars exhibit the sBD in absorption, which may indicate the presence of rather dense CEs. Six Bn stars show emission in the H$alpha$ line, so they are reclassified as Be stars. The sBD in emission appears in Be stars with $V!sin i lesssim 250$ km,s$^{-1}$, and in absorption in both Be and Bn stars with mbox{$V!sin i gtrsim 50$ km,s$^{-1}$}. Low-mass Be and Bn stars share the same region in the Hertzsprung-Russell diagram. The distributions of rotational to critical velocity ratios of Be and Bn stars corresponding to the current stellar evolutionary stage are similar, while distributions inferred for the zero-age main sequence have different skewness. We found emission in the H$alpha$ line and signs of a CE in some Bn stars, which motivated us to think that Bn and Be stars probably belong to the same population. It should be noted that some of the most massive Bn stars could display the Be phenomenon at any time. The similarities found among Be and Bn stars deserve to be more deeply pursued.
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