We show how Lynden-Bell's generalization of Mestel's method of flattened spheroids can be adapted and simplified to find biorthogonal potential- density sets for non-axisymmetric flat discs. The simplified formulae turn out to be equivalent to formulae derived by Kalnajs using a quite different method. We then use the simplified formulae to obtain two new biorthogonal potential-density sets. One is related to Gaussian discs and the other to general Toomre models, and both should be useful for stability analyses of these discs. We also give, in an appendix, a large new family of potential-density pairs described by Meijer's G-function. An interesting member of this family has density components that are infinite at the centre, yet decay exponentially at large distance.