The solution of problems in classical elastodynamics remains, for the most part, an extremely complicated and difficult task. Such problems, in addition to having all of the inherent difficulties of a boundary value problem, are, in general, subject to initial conditions as well. The work done in this area up to the present time has been chiefly in connection with special types of motion or geometric configuration. It appears that efforts to evolve a universal method of attack have not been overly successful. In regard to unified methods of approach, it is recalled that beginning with Betti’s work in 1872 [l], a vast amount of literature appeared in which the boundary value problems of static elasticity were treated by the influence function technique, similar to that used in connection with problems in potential theory. The majority of the work of this type published during the quarter-century prior to 1900 was written by Italians: Betti, Somigliana, Cerutti, Lauricella, Tedone, and others. It seems only natural that in this atmosphere someone might attempt to apply a similar procedure to the solution of problems in elastokinetics. Cerutti [2] did make such a try in 1880, and though his paper contained some inaccuracies, it perhaps served as a stimulus to others. In a rather lengthy exposition published in 1807, Tedone [3] obtained integral representations for the displacement vector in terms of the initial data and boundary displacement and traction. A more recent work by Arzhanikh [4] outlines a method of obtaining the Laplace transform of the displacement, dilatation, and rotation by use of Green’s and Neumann’s functions of potential theory. His technique is open to objection, however, because it does not yield any of the quantities explicitly, but gives an interlaced set of integral equations. Since the Laplace transformation converts the originally hyperbolic set of equations of motion into a set of elliptic equations, it would appear that the set of transformed equations might be more amenable to solution by influence