The paper by Gregory W. Horndeski discusses the construction of the most general second-order Euler-Lagrange tensors that can be derived from a Lagrangian density involving a pseudo-Riemannian metric tensor, a scalar field, and their derivatives of arbitrary order in a four-dimensional space. The author shows that these Euler-Lagrange tensors can be obtained from a Lagrangian that is at most of second order in the derivatives of the field functions. The paper begins with an introduction to the necessary mathematical preliminaries, including the definitions of the pseudo-Riemannian metric, Christoffel connection, Riemann-Christoffel curvature tensor, Ricci tensor, scalar curvature, and Einstein tensor. The main results are presented in two equations: one for the energy-momentum tensor \( E^{a b}(L) \) and another for the Lagrangian density \( E(L) \), both of which involve arbitrary functions of the scalar field and its derivatives. These equations are crucial for understanding the field equations in scalar-tensor theories, such as the Brans-Dicke theory, and are essential for further theoretical developments in this area.The paper by Gregory W. Horndeski discusses the construction of the most general second-order Euler-Lagrange tensors that can be derived from a Lagrangian density involving a pseudo-Riemannian metric tensor, a scalar field, and their derivatives of arbitrary order in a four-dimensional space. The author shows that these Euler-Lagrange tensors can be obtained from a Lagrangian that is at most of second order in the derivatives of the field functions. The paper begins with an introduction to the necessary mathematical preliminaries, including the definitions of the pseudo-Riemannian metric, Christoffel connection, Riemann-Christoffel curvature tensor, Ricci tensor, scalar curvature, and Einstein tensor. The main results are presented in two equations: one for the energy-momentum tensor \( E^{a b}(L) \) and another for the Lagrangian density \( E(L) \), both of which involve arbitrary functions of the scalar field and its derivatives. These equations are crucial for understanding the field equations in scalar-tensor theories, such as the Brans-Dicke theory, and are essential for further theoretical developments in this area.