  In this paper we have studied the two-pion exchange three-nucleon potential
$(2\pi E-3NP)$ using an approximate $SU(2) \times SU(2)$ chiral symmetry of the
strong interaction. The off-shell pion-nucleon scattering amplitudes obtained
from the Weinberg Lagangian are supplemented with contributions from the
well-known $\sigma$-term and the $\Delta(1232)$ exchange. It is the role of the
$\Delta$-resonance in $2\pi E-3NP$, which we have investigated in detail in the
framework of the Lagrangian field theory. The $\Delta$-contribution is quite
appreciable and, more significantly, it is dependent on a parameter Z which is
arbitrary but has the empirical bounds $|Z| \leq 1/2$. We find that the
$\Delta$-contribution to the important parameters of the $2\pi E-3NP$ depends
on the choice of a value for Z, although the correction to the binding energy
of triton is not expected to be very sensitive to the variation of Z within its
bounds.
