This text discusses the theory of transverse vibrations of elastic plates, focusing on the contributions of Sophie Germain and later developments by Kirchhoff and Poisson. Sophie Germain proposed a hypothesis for the forces resisting deformation in plates, leading to a partial differential equation for vibrations. However, her theory lacked proper boundary conditions. She later refined her hypothesis, allowing for the derivation of equations for vibrating plates, including those with curved surfaces. Despite experimental validation, her theory was incomplete, as it led to conclusions conflicting with reality. Kirchhoff analyzed the equilibrium and motion of elastic plates, pointing out flaws in Germain's approach. He derived a general equilibrium condition that showed the plate could not achieve equilibrium under certain conditions, contradicting Germain's theory. Poisson later developed a similar theory, but it also had limitations. Kirchhoff corrected Poisson's theory by deriving new boundary conditions, showing that plates generally do not have equilibrium positions under the given conditions. The text also presents a general equilibrium condition for elastic bodies, derived from variational principles. This condition is applied to plates, leading to partial differential equations for their vibrations. Kirchhoff's analysis shows that the equilibrium condition for plates is more general than previously thought, and that the theory must account for the curvature and deformation of the plate. The final equations derived describe the equilibrium and motion of plates under various conditions, including those with small deformations and specific boundary conditions.This text discusses the theory of transverse vibrations of elastic plates, focusing on the contributions of Sophie Germain and later developments by Kirchhoff and Poisson. Sophie Germain proposed a hypothesis for the forces resisting deformation in plates, leading to a partial differential equation for vibrations. However, her theory lacked proper boundary conditions. She later refined her hypothesis, allowing for the derivation of equations for vibrating plates, including those with curved surfaces. Despite experimental validation, her theory was incomplete, as it led to conclusions conflicting with reality. Kirchhoff analyzed the equilibrium and motion of elastic plates, pointing out flaws in Germain's approach. He derived a general equilibrium condition that showed the plate could not achieve equilibrium under certain conditions, contradicting Germain's theory. Poisson later developed a similar theory, but it also had limitations. Kirchhoff corrected Poisson's theory by deriving new boundary conditions, showing that plates generally do not have equilibrium positions under the given conditions. The text also presents a general equilibrium condition for elastic bodies, derived from variational principles. This condition is applied to plates, leading to partial differential equations for their vibrations. Kirchhoff's analysis shows that the equilibrium condition for plates is more general than previously thought, and that the theory must account for the curvature and deformation of the plate. The final equations derived describe the equilibrium and motion of plates under various conditions, including those with small deformations and specific boundary conditions.