This paper discusses the theory of elastic stability for thin shells of isotropic elastic material. The author aims to derive equations that can determine the stability under stress of such shells. The equations must account for terms that can be neglected in applications of elasticity theory when stability is not a concern. The paper uses an extended form of Hooke's Law and two methods to derive the equations: considering forces acting on an elementary volume of the substance and calculating the strain energy function to the third order of displacement coordinates. The equations are then reduced to those involving only the displacements of points on the middle surface of the shell. The boundary conditions at the faces of the shell are also discussed, and the stability of a plane plate is analyzed as an example. The final shell equations are derived, which are correct to the second order of displacement coordinates.This paper discusses the theory of elastic stability for thin shells of isotropic elastic material. The author aims to derive equations that can determine the stability under stress of such shells. The equations must account for terms that can be neglected in applications of elasticity theory when stability is not a concern. The paper uses an extended form of Hooke's Law and two methods to derive the equations: considering forces acting on an elementary volume of the substance and calculating the strain energy function to the third order of displacement coordinates. The equations are then reduced to those involving only the displacements of points on the middle surface of the shell. The boundary conditions at the faces of the shell are also discussed, and the stability of a plane plate is analyzed as an example. The final shell equations are derived, which are correct to the second order of displacement coordinates.