This paper presents the theory of elastic stability, focusing on the derivation of equations to determine the stability of thin shells under stress. The equations are based on the principles of elasticity and include terms that account for the behavior of the shell under various stresses. The paper discusses two methods for deriving these equations: one based on the equilibrium of an elementary volume and another based on the energy of strain. The energy method is preferred as it simplifies the process by using the strain energy function. The paper also addresses the reduction of equations involving displacements of all points of the shell to equations involving only the displacements of points on the middle surface. This is similar to the approach used in the Theory of Thin Shells. The assumptions used in this reduction are such that they allow for the simplification of the equations, even though they may not be applicable to all cases. The paper then focuses on the strain energy function, which is a quadratic function of the components of strain. The energy of strain is calculated using an extended form of Hooke's Law, which accounts for the effects of strain on the material. The paper also discusses the conditions for the stability of a plane plate, showing how the equations can be derived and applied to determine the critical values of external forces that lead to instability. The paper concludes with the derivation of the strain energy function for a thin shell, which includes terms up to the third order of displacement coordinates. The final expression for the strain energy function is given, which includes terms related to the elastic constants and the displacements of the shell. The paper also discusses the boundary conditions and the reduction of the strain energy function to terms involving only the displacements of the middle surface. The final equations for the stability of the shell are presented, which are used to determine the critical values of external forces that lead to instability. The paper also discusses the application of these equations to the stability of a tubular strut, showing how the results can be checked against known formulas.This paper presents the theory of elastic stability, focusing on the derivation of equations to determine the stability of thin shells under stress. The equations are based on the principles of elasticity and include terms that account for the behavior of the shell under various stresses. The paper discusses two methods for deriving these equations: one based on the equilibrium of an elementary volume and another based on the energy of strain. The energy method is preferred as it simplifies the process by using the strain energy function. The paper also addresses the reduction of equations involving displacements of all points of the shell to equations involving only the displacements of points on the middle surface. This is similar to the approach used in the Theory of Thin Shells. The assumptions used in this reduction are such that they allow for the simplification of the equations, even though they may not be applicable to all cases. The paper then focuses on the strain energy function, which is a quadratic function of the components of strain. The energy of strain is calculated using an extended form of Hooke's Law, which accounts for the effects of strain on the material. The paper also discusses the conditions for the stability of a plane plate, showing how the equations can be derived and applied to determine the critical values of external forces that lead to instability. The paper concludes with the derivation of the strain energy function for a thin shell, which includes terms up to the third order of displacement coordinates. The final expression for the strain energy function is given, which includes terms related to the elastic constants and the displacements of the shell. The paper also discusses the boundary conditions and the reduction of the strain energy function to terms involving only the displacements of the middle surface. The final equations for the stability of the shell are presented, which are used to determine the critical values of external forces that lead to instability. The paper also discusses the application of these equations to the stability of a tubular strut, showing how the results can be checked against known formulas.