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A double transform method is used to solve the problem of determining the elastic strain in a semi-infinite cylindrical bar with a stress free lateral surface, subject to the end conditions that the stress applied normally to the end is uniform and has a step function time dependence and that the radial displacement at the end is always zero. The exact solution appears as a sum of Fourier integrals whose integrands have the form of Pochhammer-Chree waves. These integrals cannot be evaluated in general by simple means, but asymptotic solutions have been obtained which are valid for large distances of travel. The theoretical predictions are compared with the results of experiment in a companion report.

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The variational principle for the energy is used to derive integrodifferential equations for the two-body functions which, when combined in the product form, yield the "best" independent-pair wave function for the a particle. A practicable iteration procedure for finding approximate solutions to these equations is developed and is used to obtain an approximate ground-state wave function, for an example, four-bodyHamiltonian for which the potential is central. Improvements in the iteration procedure are described which make it a feasible method for solving the integrodifferential equations even when tensor forces are included.

Helmholtz' decomposition theorem for vector fields is presented usually with too strong restrictions on the fields. Based on the work of Blumenthal of 1905 it is shown that the decomposition of vector fields is not only possible for asymptotically weakly decreasing vector fields, but even for vector fields, which asymptotically increase sublinearly. Use is made of a regularization of the Green's function and the mathematics of the proof is formulated as simply as possible. We also show a few examples for the decomposition of vector fields including the electric dipole radiation.