  In a space of 4-dimensions, I will examine constrained variational problems
in which the Lagrangian, and constraint scalar density, are concomitants of a
(pseudo-Riemannian) metric tensor and its first two derivatives. The Lagrange
multiplier for these constrained extremal problems will be a scalar field. For
suitable choices of the Lagrangian, and constraint, we can obtain
Euler-Lagrange equations which are second order in the scalar field and third
order in the metric tensor. The effect of disformal transformations on the
constraint Lagrangians, and their generalizations, is examined. This will yield
other second order scalar-tensor Lagrangians which yield field equations which
are at most of third order. No attempt is made to construct all possible third
order scalar-tensor Euler-Lagrange equations in a 4-space, although nine
classes of such field equations are presented. Two of these classes admit
subclasses which yield conformally invariant field equations. A few remarks on
scalar-tensor-connection theories are also presented.
