The problem of the buckling of a simply supported thin, elastic, cylindrical panel subjected to a uniform axial compression is studied by constructive methods. In the case of “narrow” panels we obtain near the classical buckling load $\lambda _1 $ a description of all stable and unstable buckled states which branch from the unbuckled state at $\lambda _1 $. The method also yields useful information on the asymptotic form of the buckled states near $\lambda _1 $, the exchange of stabilities at $\lambda _1 $, and whether the buckling at the ends of a narrow panel is “inward or outward”.