[For the entire collection see Zbl 0539.00015.]
It was observed by S. Zaremba [Bull. Int. Acad. Sci. Cracovie, 147- 196 (1907)] that the Dirichlet problem could be solved by the use of complete orthonormal sets of harmonic functions. In what follows, we will explain how his ideas can be used to symbolically calculate an approximate solution of the Dirichlet problem in the form of a harmonic polynomial P(x,y) by orthonormalizing sets of harmonic polynomials. In fact, our approximate solution will be a harmonic polynomial and will provide a value of the solution at every point of a domain \(D\subset {\mathbb{R}}^ 2\). To find this solution explicitly, we will make extensive use of the symbolic manipulation language FORMAC which makes Zaremba’s solution possible in a truly computational sense.