  In a four-dimensional space I shall construct all of the conformally
invariant, scalar-vector-tensor field theories that are consistent with
conservation of charge, and flat space compatible. By the last assumption I
mean that the Lagrangian of the theory in question is well defined and
differentiable when evaluated for either a flat metric tensor, (and) or
constant scalar field, (and) or vanishing vector potential. The Lagrangian of
any such field theory can be chosen to be a linear combination of six
conformally invariant scalar-vector-tensor Lagrangians, with the coefficients
being scalar functions of the scalar field. Five of these generating
Lagrangians are at most of second-order, while the sixth one is 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 scalar-vector-tensor field theories that are consistent with
conservation of charge, and flat space compatible, can be obtained from a
second-order Lagrangian. The vector equation of any such theory is at most of
second-order, and is an extension of Maxwell's equations, incorporating two
other first-order terms that vanish when the scalar field is constant. Hence in
regions where the scalar field is constant, the vector equation reduces to
Maxwell's.
