In a recent investigation, a method was given for the approximate solution of certain boundary-value problems. This method lends itself well to the use of the electronic digital computer and is extended here to investigate the eigenvalue problems of the buckling under two-dimensional hydrostatic loading and the vibration of thin plates. The two-dimensional hydrostatic buckling loads of clamped square and equilateral-triangular plates are found by this method, the values agreeing well with the results obtainable by other methods where these results are known.