This paper is concerned with the determination of the distribution of stress in an infinite elastic solid when time-dependent body forces act upon certain regions of the solid. It is assumed throughout that the strains are small. In section 2 a general solution of the equations of motion for any distribution of body forces is derived by the use of four-dimensional Fourier transforms, and from that is derived the general solution for an isotropic solid (section 3).From the latter solution are deduced the general solution of the statical problem (section 4) and the two-dimensional problem (section 5). The solution of the equations of motion in the case in which the distribution of body forces is symmetrical about an axis is derived in section 6. The remainder of the paper consists in deducing the solution of special problems from these general solutions. In section section 7 to 13 some typical two-dimensional problems are considered and exact analytical expressions found for the components of the stress tensor. In section section 14 to 16 examples are given of the use of the general non-symmetrical three-dimensional solution derived in section 3, and in section section 17 to 19 examples are given to illustrate the use of the general solution of the axially symmetrical problem. A certain amount of numerical work (presented in graphical form) is quoted to give some idea of the physical nature of the solutions.