The problem to be discussed is as follows. Suppose a mathematical model for a given physical problem results in a self-adjoint eigenvalue problem of the form $$\begin{gathered}w(4) + (Aw(1))(1) + Bw - \lambda w = 0 \hfill \\\sum\limits_{i = 1}^4 {\alpha _{ij} w(i - 1)(0) = 0 = } \sum\limits_{i = 1}^4 {\beta _{ij} w(i - 1)(1),j = 1,2.} \hfill \\\end{gathered}$$ Suppose A, B and possibly αij, βij, i=1,...,4, j=1,2 are unknown but eigenvalues λi, i=1,2,... are known. A constructive technique for finding the unknown coefficients is presented. Additional data which can be required known are two normalization constants for each of the eigenfunctions.