  The existence of a nuclear electric octupole moment (EOM) requires both
parity and time invariance violation. The EOMs of odd $Z$ nuclei that are
induced by a particular T- and P-odd interaction are calculated. We compare
such octupole moments with the collective EOMs that can occur in nuclei having
a static octupole deformation. A nuclear EOM can induce a parity and time
invariance violating atomic electric dipole moment, and the magnitude of this
effect is calculated. The contribution of a nuclear EOM to such a dipole moment
is found, in most cases, to be smaller than that of other mechanisms of atomic
electric dipole moment production.
