[en] A method proposed earlier by Aguilera, Moshinsky, and Kramer, for adapting a system of translationally invariant four-particle harmonic oscillator functions to the symmetry of the permutation group S (4), is applied hyperspherical harmonic functions depending on three relative vectors. Except for a few cases in which diagonalization of matrices is required, the method gives closed formulas for orthonormal sets of harmonic functions with good permutational symmetry. The matrix elements of S (4) permutations with respect to the harmonic functions are obtained