The surface rheological theory presented in the first part of this series for the planar fluid surface is extended to curved fluid surfaces with curvature approaching that which is observed in microemulsion systems. Both the surface excess linear momentum equation and the moment of linear momentum equation are developed. The balances are for the first time generalized to include large surface curvature stresses, introducing new first and second moments of surface excess shear and dilatational viscosity. The classical “zeroth order” surface excess shear and dilatational viscosities are shown to be curvature dependent. The curvature dependency of each zeroth moment surface viscosity coefficient is shown completely defined by the respective first and second moment viscosity coefficients.