The flat problem of elasticity theory for rectangle, two sides of which are fastened and the other two sides are under the influence of external tangent loadings is considered. For simplicity it is assumed that external tangent loadings are absent.
   The task is solved by means of Furie method. The solution is presented in a form of the sum of the two Furie rows, each of them contains four groups of free coefficient of integration. A part of free coefficients is precisely determined, but for the determination of remain coefficients of Furie rows the collection of four infinite systems of linear algebraic equations is obtained. It is proved that infinite systems are quite regular and their free members are tending to zero. Consequently, infinite systems could be solved either through the method of progressive approximation or the method of reduction.