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A technique is developed for constructing the solution of ${\Delta }^{2}u=F$ in $R=\left\{\right(x,y):|x|\<a,|y|\<b\}$, subject to boundary conditions $u=\phi$, $\frac{\partial u}{\partial n}=\psi$ on $\partial R$. The problem is reduced to that of finding the orthogonal projection $Pw$ of $w$ in $L^2\left(R\right)$ onto the subspace $\mathbf{H}$ of square integrable functions harmonic in $\mathbf{R}$. This problem is solved by decomposition $\mathbf{H}$ into the closed direct (not orthogonal) sum of two subspaces ${\mathbf{H}}^{\left(1\right)},{\mathbf{H}}^{\left(2\right)}$ for which complete orthogonal bases are known. $P$ is expressed in terms of the projections $P^{\left(1\right)}$, $P^{\left(2\right)}$ of $L^2\left(R\right)$ onto ${\mathbf{H}}^{\left(1\right)}$, ${\mathbf{H}}^{\left(2\right)}$ respectively. The resulting construction...