A simple method is given which is suitable for the approximate analysis of certain boundary-value problems, including, for example, the small deflections of clamped plates and the torsion of prismatic bars. The analysis is particularly simple and lends itself well to the use of the digital computer. The method is applied here to four problems, the uniformly loaded, clamped square, and equilateral-triangle plates, and the torsion of bars of square and hexagonal cross section. The results agree well with the exact solutions, where these are known.