  The Horndeski theory is known as the most general scalar-tensor theory with
second-order field equations. In this paper, we explore the bi-scalar extension
of the Horndeski theory. Following Horndeski's approach, we determine all the
possible terms appearing in the second-order field equations of the
bi-scalar-tensor theory. We compare the field equations with those of the
generalized multi-Galileons, and confirm that our theory contains new terms
that are not included in the latter theory. We also discuss the construction of
the Lagrangian leading to our most general field equations.
