  We derive four-dimensional relativistic three-body equations for the case of
a field theory with a three-point interaction vertex. These equations describe
the coupled 2->2, 2->3, and 3->3 processes, and provide the means of
calculating the kernel of the 2->2 Bethe-Salpeter equation. Our equations
differ from all previous formulations in two essential ways. Firstly, we have
overcome the overcounting problems inherent in earlier works. Secondly, we have
retained all possible two-body forces when one particle is a spectator. In this
respect, we show how it is necessary to also retain certain three-body forces
as these can give rise to (previously overlooked) two-body forces when used in
a 2->3 process. The revealing of such hidden two-body forces gives rise to a
further novel feature of our equations, namely, to the appearance of a number
of subtraction terms. In the case of the piNN system, for example, the NN
potential involves a subtraction term where two pions, exchanged between the
nucleons, interact with each other through the pi-pi t-matrix. The necessity of
an input pi-pi interaction is surprising and contrasts markedly with the
corresponding three-dimensional description of the piNN system where no such
interaction explicitly appears. This illustrates the somewhat unexpected result
that the four-dimensional equations differ from the three-dimensional ones even
at the operator level.
