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The natural boundary conditions for corners of a certain type of two-dimensional problem are derived from the variational principle. A plate problem formulated by the principle of minimum potential energy belongs to this type of problem. The physical problem requires that the first variation of the total potential energy vanish, which involves a functional integral with second partial derivatives. The first variation of a functional with second partial derivatives for a general 2-D problem is studied for a closed region bounded by piecewise smooth curves. A line integral along the edge boundaries is obtained from the integration by parts of the integral. Natural boundary conditions along smooth edges are obtained directly from the line integral, and the natural boundary condition at a corner is obtained by considering the discontinuity of the slopes of the two edges at the corner. Curvilinear coordinates are considered. However, the boundary curves may or may not coincide with the coordinate lines.