Two examples are considered of the direct problem of transonic gas flow in the formulation of F. I. Frankl': the problem of flow through a Laval nozzle with nearly parallel walls, and the problem of flow past a symmetric wedge-like profile in a sonic gas stream. Calculation of the flow in the first approximation is reduced to a boundary-value problem for a second-order equation of mixed type. The boundary-value problem is in turn transformed to a singular integral equation with kernel of Cauchy type. The solution of the equation is sought by a method of successive approximations. A condition is found that permits determination of the flux in the case of flow through a Laval nozzle, or the coefficient for the singular solution in the case of sonic gas flow past a profile. The presence of nonlinear terms in the equations of transonic gas flow leads to great difficulties in the solution of the direct problem (the problem of finding the flow in a region having given boundaries). In the case of plane parallel flow it is possible to transform the nonlinear equations in the flow plane into linear ones in the plane of the velocity hodograph. However the boundary-value problem can be posed in the hodograph plane only for a limited number of flows, with boundaries on which a relation between the velocity components is known in advance. A. A. Nikol'skii [1] proposed a method of solving the boundary-value problem with boundaries that differ only slightly from those for which the transformation into the hodograph plane is known. As a result one obtains a boundary-value problem with linear boundary conditions for a linear second-order differential equation of mixed type. The method of A. A. Nikol'skii was further developed in the work of F. I. Frankl' [2, 3], where it was applied to the solution of the direct problem of the Laval nozzle. It was shown that the flux through a nozzle with given walls is essentially undetermined, and can be prescribed in an arbitrary way. In his subsequent work [4, 5] F. I. Frankl' discovered an undetermined coefficient also in the problem of sonic gas flow past a profile. In this case the solution is determined only to within an arbitrary multiplicative constant at the point of hodograph plane corresponding to the region infinitely remote from the body. In these same papers the premise was advanced that the solution of the direct problem of transonic flow must satisfy some supplementary conditions, having a physical basis, which serve for finding the undetermined parameters in the problem. The question of uniqueness of the flux through a given Laval nozzle was considered also in [6], in which uniqueness of flux was proved under the condition of conservation of the asymptotic type of flow at the center of the nozzle.