  In a four-dimensional space, I shall construct all of the conformally
invariant scalar-tensor field theories, which are flat space compatible; i.e.,
well-defined and differentiable when evaluated for a flat metric tensor and
constant scalar field. It will be shown that all such field theories must be at
most of fourth-order in the derivatives of the field variables. The Lagrangian
of any such field theory can be chosen to be a linear combination of four
conformally invariant scalar-tensor Lagrangians, with the coefficients being
functions of the scalar field. Three of these "generating" Lagrangians are of
second-order, while one of of third-order. However, the third-order Lagrangian
differs from a non-conformally invariant second-order Lagrangian by a
divergence. Consequently, all of the conformally invariant, flat space
compatible, scalar-tensor field theories, can be obtained from a second-order
Lagrangian.
