  In this paper I shall consider field theories in a space of four-dimensions
which have field variables consisting of the components of a metric tensor and
scalar field. The field equations of these scalar-tensor field theories will be
derivable from a variational principle using a Lagrange scalar density which is
a concomitant of the field variables and their derivatives of arbitrary, but
finite, order. I shall consider biconformal transformations of the field
variables, which are conformal transformations which affect both the metric
tensor and scalar field. A necessary and sufficient condition will be developed
to determine when the Euler-Lagrange tensor densities are biconformally
invariant. This condition will be employed to construct all of the second-order
biconformally invariant scalar-tensor field theories in a space of
four-dimensions. It turns out that the field equations of these theories can be
derived from a linear combination of (at most) two second-order Lagrangians,
with the coefficients in that linear combination being real constants.
