UDC 517.9 w Introduction In the linear theory of thin wing (see [1-3] etc.), the Laplace equation for the velocity potential (I) is supplied with the usual boundary conditions as well as with the requirement of boundedness of the gradient grad (I) of the velocity potential on the trailing edge F of the wing. The possibility of satisfying this condition is associated with partial indefiniteness of the Dirichlet data on a part of the boundary, namely, they involve the unknown flow circulation whose values coincide with the trace of the potential on the line (edge) F. The requirement of boundedness of grad ~, which is attributed to Chaplygin, Zhukovskii, and Kutta, is mentioned in [4-6] and stated in [7]. In the two-dimensional situation, which has been studied comprehensively (see [1-9] etc.), F degenerates into a point and the operator of the problem with the Chaplygin-Zhukov skii-Kutta condition inherits the Fredholm property from the operator of the usual boundary value problem. As to the three-dimensional case, here the additional restrictions and arbitrariness do not fit in finite-dimensional spaces and thus can result in loss of the Fredholm property. So far no exhaustive mathematical investigation on the correct statement of the Chaplygin-Zhukov skii-Kutta conditions has been carried out (the author is aware of only one publication [10] where an attempt was made to use Sobolev spaces for this purpose, but the definitive answer was not given). In the present article we prove the Fredholm theorem for a problem that models the actual one. Specifically, we consider the problem in a bounded domain (rather than in R 3) and prescribe the Chal~lyginZhukovskii-Kutta conditions on a smooth closed curve F (the trailing edge of the wing has corner points and ends). The problem is described completely in w The function space in which the operator of the problem has the Fredholm property consists of functions with asymptotics of a certain form, and the Chaplygin-Zhukov skii-Kutta condition is restated as the requirement that the coefficient in the asymptotic term inducing the square root singularities vanish. All this, including the statement of the theorem on the Fredholm property, is contained in w167 3 and 4. The proof of the theorem is carried out as follows: first, a model problem in R~ is investigated in w and then the procedure of freezing the coefficients is applied in w All constructions are devised for concrete equations, and their generalization is not a purpose of this work (conditions on submanifolds of the boundary were studied for some situations in [11-13], etc.). In this article, we use the results of [14-21] in the theory of boundary value problems in domains with piecewise smooth boundary; they are collected in the book [21], to which we give exact references. I express my deep gratitude to Prof. Meister and Prof. Hinder for useful discussions of the subject of this paper. w Statement of the Problem