The Euler-Lagrange equations corresponding to a Lagrange density which is a function of gij and its first two derivatives are investigated. In general these equations will be of fourth order in gij. Necessary and sufficient conditions for these Euler-Lagrange equations to be of second order are obtained and it is shown that in a four-dimensional space the Einstein field equations (with cosmological term) are the only permissible second order Euler-Lagrange equations. This result is false in a space of higher dimension. Furthermore, the only permissible third order equation in the four-dimensional case is exhibited.