Like Fig. 1, but for the central column density (left panel) and mean column density (right panel) through an isothermal self-gravitating cylinder (black) and sphere (grey), as a function of mass fraction f. The dashed line in the left-hand plot is the approximation given in Eq. (19). The dashed lines in the right-hand plot echo the central column densities from the left-hand plot for comparison. The horizontal dotted lines give the values achieved by a critical stable sphere. The values for a cylinder with that overpressure, mass fraction 0.733, are shown as filled black circles.