The application of ultrasonic methods to the determination of the elastic constants of solids is considered in some detail. It is shown that a rotating plate technique in which ultrasonic transmission is plotted as a function of the angle of incidence of the waves allows determination of the velocities of dilatation and shear waves in the plate. From these data, Poisson's ratio and the mechanical moduli may be determined. Details of an apparatus for making such measurements are given. The elastic constants of several metals have been measured with this equipment and the values obtained are shown to be in agreement with previous published data. In addition, measurements of a number of thermoplastic and thermosetting resins have been made successfully. It has been found that in the case of Melmac Resin 26–8B there is no variation in the elastic constants as a function of cure time.