This section discusses the theories of equilibrium and motion of elastic plates, focusing on the contributions of Sophie Germain and Poisson. Sophie Germain's initial work, published in 1811, introduced a hypothesis about the forces that cause a plate to deform and derived a partial differential equation for the vibrations. However, her calculations contained a mistake, and Lagrange corrected it to obtain the correct equation. Germain later derived boundary conditions for rectangular plates and compared her theoretical results with experimental observations, finding agreement. In her third paper, she extended her hypothesis to include the theory of plate vibrations in naturally curved shapes, such as cylinders. Despite the validation of her theory through experiments, it was later shown that her conclusions were inconsistent with reality. The author demonstrates this by analyzing a plate in its natural state, where the conditions for equilibrium derived from Germain's hypothesis do not hold, leading to the conclusion that such a plate generally does not achieve equilibrium. Poisson's theory, developed in his famous paper "On the Equilibrium and Motion of Elastic Bodies," also requires correction. The author derives a more general equation for the equilibrium of elastic bodies and shows that Poisson's three boundary conditions cannot all be satisfied simultaneously, leading to the conclusion that Poisson's theory also fails to describe the equilibrium of a plate in general. The author aims to provide a corrected theory and will derive the necessary boundary conditions to replace Poisson's three conditions.This section discusses the theories of equilibrium and motion of elastic plates, focusing on the contributions of Sophie Germain and Poisson. Sophie Germain's initial work, published in 1811, introduced a hypothesis about the forces that cause a plate to deform and derived a partial differential equation for the vibrations. However, her calculations contained a mistake, and Lagrange corrected it to obtain the correct equation. Germain later derived boundary conditions for rectangular plates and compared her theoretical results with experimental observations, finding agreement. In her third paper, she extended her hypothesis to include the theory of plate vibrations in naturally curved shapes, such as cylinders. Despite the validation of her theory through experiments, it was later shown that her conclusions were inconsistent with reality. The author demonstrates this by analyzing a plate in its natural state, where the conditions for equilibrium derived from Germain's hypothesis do not hold, leading to the conclusion that such a plate generally does not achieve equilibrium. Poisson's theory, developed in his famous paper "On the Equilibrium and Motion of Elastic Bodies," also requires correction. The author derives a more general equation for the equilibrium of elastic bodies and shows that Poisson's three boundary conditions cannot all be satisfied simultaneously, leading to the conclusion that Poisson's theory also fails to describe the equilibrium of a plate in general. The author aims to provide a corrected theory and will derive the necessary boundary conditions to replace Poisson's three conditions.