Observations indicate that interstellar molecular clouds can often be approximated by rotating (polytropic) cylinders. The two limiting configurations—the homogeneous and the isothermal cylinder—are used to obtain analytical solutions valid, at least qualitatively, for the whole range 0 ≤ n ≤ ∞ of permitted polytropic indices. The radius of critically rotating, infinitely long polytropic cylinders has a flat minimum at polytropic index n = 0.854, becoming infinite in the two limiting cases n = 0 and ∞. The ratio between the radii of critically rotating and nonrotating polytropic cylinders decreases strongly from ∞ if n = 0 to 1.52 if n = 2.04, afterward increasing slowly to exp(1/2) = 1.65 if n = ∞ . Natural polytropic variables are presented in the Appendix.