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. Many problems in Engineering and Physics can be reduced to boundary value problems, i.e. to problems involving the determination of a function which satisfies a given partial differential equation inside a domain and assumes perscribed values on the boundary of this domain. Despite the fact that the solutions of many such problems have been known "in principle," their actual evaluation has required such a great amount of computation that it has only been possible to carry out calculations of this kind in a few simple cases. The creation of modern computational devices such as punch-card machines, IBM-Harvard and Bell Telephone Laboratory machines, the ENIAC computer, etc., which can carry out rather extensive computations automatically, has changed the picture completely. However, these machines exist only to implement theoretical methods, and the reduction of these methods to a form whereby the machine can "take hold" represents a problem in itself. In previous publications the author developed certain theoretical methods (the "method of orthogonal functions" and the "method of particular solutions") for solving boundary-value problems. The present paper illustrates the application of orthogonal functions to the solution of Laplace's equation (d2</>/dx2) + (d2</>/d;y2) =0 through the use of punch-card machines. 1. Formulation of a problem in elasticity. The present paper is concerned with a method of solving the torsion problem for a bar of uniform cross-section. Let x, y, Z denote rectangular coordinates, the axis of Z being perpendicular to the cross-section of the beam.1 According to Saint Venant the components u, v, w of the displacement vector are given by the expressions

Punch-card machine methods applied to the solution of the torsion problem