  The anomalous magnetic field dependence of dielectric properties of
insulating glasses in the temperature interval $10mK<T<50mK$ is considered. In
this temperature range, the dielectric permittivity is defined by the resonant
contribution of tunneling systems. The external magnetic field regulates
nuclear spins of tunneling atoms. This regulation suppresses a nuclear
quadrupole interaction of these spins with lattice and, thus, affects the
dielectric response of tunneling systems. It is demonstrated that in the
absence of an external magnetic field the nuclear quadrupole interaction $b$
results in the correction to the permittivity $\delta\chi\sim| b| /T$ in the
temperature range of interest. An application of a magnetic field results in a
sharp increase of this correction approximately by a factor of two when the
Zeeman splitting $m$ approaches the order of $| b| $. Further increase of the
magnetic field results in a relatively smooth decrease in the correction until
the Zeeman splitting approaches the temperature. This smooth dependence results
from tunneling accompanied by a change of the nuclear spin projection. As the
magnetic field surpasses the temperature, the correction vanishes. The results
obtained in this paper are compared with experiment. A new mechanism of the low
temperature nuclear spin-lattice relaxation in glasses is considered.
