This paper discusses the theory of waves in an elastic plate, building upon previous work by Pochhammer, Rayleigh, and the author. The main objective is to determine the period-equation for the vibrations of an infinite plate, which is essential for understanding the modes of vibration. The paper considers both symmetrical and asymmetrical vibrations, and derives the wave-velocity for different types of waves, including Rayleigh waves. The motion is considered in two dimensions, with the plate's thickness denoted by 2f. The stress-strain equations are given, and the solution is expressed in terms of two functions φ and ψ, which satisfy wave equations. Assuming a time-factor $ e^{i\sigma t} $, the wave equations are transformed into Helmholtz equations. The period-equation is derived, which relates the wave-length to the wave-velocity. For long waves, the period-equation simplifies to a form that agrees with the ordinary theory of vibrations of thin rods or plates. For short waves, the period-equation approaches the form of the Rayleigh wave equation. The paper also considers the case of an incompressible material, where the wave-velocity is determined by the material's properties. The results are presented in tables, showing the wave-length, wave-velocity, and other parameters for different modes of vibration. The paper also discusses the influence of compressibility on the wave-velocity and the behavior of different modes of vibration. The paper concludes with a discussion of the physical significance of the results, noting that the wave-velocity is limited by the material's properties, and that the concept of wave-velocity is not strictly applicable to all types of waves. The results are summarized in tables and graphs, showing the relationship between wave-length and wave-velocity for different modes of vibration.This paper discusses the theory of waves in an elastic plate, building upon previous work by Pochhammer, Rayleigh, and the author. The main objective is to determine the period-equation for the vibrations of an infinite plate, which is essential for understanding the modes of vibration. The paper considers both symmetrical and asymmetrical vibrations, and derives the wave-velocity for different types of waves, including Rayleigh waves. The motion is considered in two dimensions, with the plate's thickness denoted by 2f. The stress-strain equations are given, and the solution is expressed in terms of two functions φ and ψ, which satisfy wave equations. Assuming a time-factor $ e^{i\sigma t} $, the wave equations are transformed into Helmholtz equations. The period-equation is derived, which relates the wave-length to the wave-velocity. For long waves, the period-equation simplifies to a form that agrees with the ordinary theory of vibrations of thin rods or plates. For short waves, the period-equation approaches the form of the Rayleigh wave equation. The paper also considers the case of an incompressible material, where the wave-velocity is determined by the material's properties. The results are presented in tables, showing the wave-length, wave-velocity, and other parameters for different modes of vibration. The paper also discusses the influence of compressibility on the wave-velocity and the behavior of different modes of vibration. The paper concludes with a discussion of the physical significance of the results, noting that the wave-velocity is limited by the material's properties, and that the concept of wave-velocity is not strictly applicable to all types of waves. The results are summarized in tables and graphs, showing the relationship between wave-length and wave-velocity for different modes of vibration.