This paper by Gregory Walter Horndeski discusses second-order scalar-tensor field equations in a four-dimensional space. The paper considers Lagrange scalar densities that are concomitants of a pseudo-Riemannian metric-tensor, a scalar field, and their derivatives of arbitrary order. The most general second-order Euler-Lagrange tensors derivable from such a Lagrangian in a four-dimensional space are constructed, and it is shown that these Euler-Lagrange tensors may be obtained from a Lagrangian which is at most of second order in the derivatives of the field functions. The paper is based on a real, four-dimensional, $ C^{\infty} $ differentiable manifold M. It assumes that all field functions are defined globally, but the work is purely local. A pseudo-Riemannian metric for M is a $ C^{\infty} $ symmetric (0, 2) tensor field on M that associates a non-degenerate, symmetric bilinear form to each fibre of the tangent bundle of M. The components of the metric are denoted by $ g_{ij} $, where Latin indices run from 1 to 4. The coefficients of the Christoffel connection determined by $ g_{ij} $ are given by a specific formula. The components of the Riemann-Christoffel curvature tensor are defined by a specific equation. The components of the Ricci tensor, scalar curvature, and Einstein tensor are also defined. The vacuum field equations of most scalar-tensor field theories are usually assumed to be the Euler-Lagrange equations corresponding to some suitably chosen Lagrange scalar density which is a concomitant of a pseudo-Riemannian metric tensor. A scalar field and their derivatives are considered, for example the Brans-Dicke (1961) field theory. It is usually demanded that the field equations be at most of second-order in the derivatives of both sets of field functions. Horndeski and Lovelock (1972) have shown that in a four-dimensional space the most general second-order Euler-Lagrange equations which can be derived from a Lagrange scalar density of the form $ L = L(g_{ij}; g_{ij,h}; g_{ij,hk}; \phi; \phi_{,i}) $ are given by a complex formula.This paper by Gregory Walter Horndeski discusses second-order scalar-tensor field equations in a four-dimensional space. The paper considers Lagrange scalar densities that are concomitants of a pseudo-Riemannian metric-tensor, a scalar field, and their derivatives of arbitrary order. The most general second-order Euler-Lagrange tensors derivable from such a Lagrangian in a four-dimensional space are constructed, and it is shown that these Euler-Lagrange tensors may be obtained from a Lagrangian which is at most of second order in the derivatives of the field functions. The paper is based on a real, four-dimensional, $ C^{\infty} $ differentiable manifold M. It assumes that all field functions are defined globally, but the work is purely local. A pseudo-Riemannian metric for M is a $ C^{\infty} $ symmetric (0, 2) tensor field on M that associates a non-degenerate, symmetric bilinear form to each fibre of the tangent bundle of M. The components of the metric are denoted by $ g_{ij} $, where Latin indices run from 1 to 4. The coefficients of the Christoffel connection determined by $ g_{ij} $ are given by a specific formula. The components of the Riemann-Christoffel curvature tensor are defined by a specific equation. The components of the Ricci tensor, scalar curvature, and Einstein tensor are also defined. The vacuum field equations of most scalar-tensor field theories are usually assumed to be the Euler-Lagrange equations corresponding to some suitably chosen Lagrange scalar density which is a concomitant of a pseudo-Riemannian metric tensor. A scalar field and their derivatives are considered, for example the Brans-Dicke (1961) field theory. It is usually demanded that the field equations be at most of second-order in the derivatives of both sets of field functions. Horndeski and Lovelock (1972) have shown that in a four-dimensional space the most general second-order Euler-Lagrange equations which can be derived from a Lagrange scalar density of the form $ L = L(g_{ij}; g_{ij,h}; g_{ij,hk}; \phi; \phi_{,i}) $ are given by a complex formula.