Abstract: In this paper, an exact integral equation based on the transonic small perturbation equation is derived for two-dimensional transonic small perturbation flows over aerofoils. In the process of the derivation, a new idea is given to deal with the singular point, i.e., a closed curve Co of arbitrary shape vanishing in the limit, is used to exclude the singular point Q, in result, an integral equation without infinite singularity is obtained which is convenient for calculation.The paper proposses a new method to tract the problem of the leading edges of round-nosed aerofoils in the transonic flows. Results of numerical calculation show that the influences of the round noses on the flows are limited in small regions about them.The inversion formula given by Nixon is proved to be valid also for supercritical small perturbation flows with shocks.The method of artificial viscosity in the integral equation methods, first proposed by Sochdev and Lobo, has been improved in the paper.Finally, results of numerical computation are given for NACA 0012 aerofoil in lifting and non-lifting flows with and without shocks, and are compared with some other numerical results, which shows that the present method is a good one for numerical accuracy and computational economy.