  Neutron transmission experiments can offer a new type of highly sensitive
search for time-reversal invariance violating (TRIV) effects in nucleon-nucleon
interactions via the same enhancement mechanism observed for large parity
violating (PV) effects in neutron-induced compound nuclear processes. In these
compound processes, the TRIV cross-section is given as the product of the PV
cross-section, a spin-factor $\kappa$, and a ratio of TRIV and PV matrix
elements. We determined $\kappa$ to be $0.59\pm0.05$ for $^{139}$La+$n$ using
both $(n, \gamma)$ spectroscopy and ($\vec{n}+^{139}\vec{\rm La}$)
transmission. This result quantifies for the first time the high sensitivity of
the $^{139}$La 0.75~eV $p$-wave resonance in a future search for P-odd/T-odd
interactions in ($\vec{n}+^{139}\vec{\rm La}$) forward transmission.
