  The empirical Bath's law is derived from the magnitude-difference statistical
distribution of earthquake pairs. The pair distribution related to earthquake
correlations is presented. The single-event distribution of dynamically
correlated earthquakes is derived, by means of the geometric-growth model of
energy accumulation in the focal region. The dynamical correlations may
account, at least partially, for the roll-off effect in the Gutenberg-Richter
distributions. The seismic activity which accompanies a main shock, including
both the aftershocks and the foreshocks, can be viewed as fluctuations in
magnitude. The extension of the magnitude difference to negative values leads
to a vanishing mean value of the fluctuations and to the standard deviation as
a measure of these fluctuations. It is suggested that the standard deviation of
the magnitude difference is the average difference in magnitude between the
main shock and its largest aftershock (foreshock), thus providing an insight
into the nature and the origin of the Bath's law. It is shown that
moderate-magnitude doublets may be viewed as Bath partners. Deterministic
time-magnitude correlations of the accompanying seismic activity are also
presented.
