Consider an isolated chamber divided into two parts by a membrane possessing surface tension $\sigma$ and area $A$. Let the two parts have equal temperature $T$, and let them have pressures $p_1, p_2$ and volumes $V_1, V_2$. Each part can be treated as a simple compressible system. (i) By considering a fluctuation in which a ripple appears on the membrane, changing its area without affecting $V_1$ and $V_2$, show that if the volumes are fixed then in thermal equilibrium the membrane adopts the shape of least area consistent with the fixed volumes. (ii) By considering a fluctuation in which $V_1$ grows and $V_2$ shrinks and $A$ also consequently changes, show that the equilibrium condition $\mathrm{d} S=0$ gives $\sigma=\left(p_1-p_2\right) \mathrm{d} V_1 / \mathrm{d} A$ (cf. equation (14.10)). Hence relate the pressure difference at any point on the surface to the local radius of curvature of the surface.