  We have written a new equation to study the statistics of earthquake
distributions. We call this equation "the generalized logistic equation". The
Gutenberg-Richter frequency-magnitude formula was derived from the solution of
the generalized logistic equation as an asymptotic case in approximation of
large magnitudes. To illustrate how the found solution of the generalized
logistic equation works, it was used to approximate the observed cumulative
distribution of earthquakes in four different geological provinces: the Central
Atlantic (40N-25N, 5W-35W), Canary Islands, Magellan Mountains (20N-9S,
148E-170E), and the Sea of Japan. This approximation showed the excellent fit
between the theoretical curves and observed data for earthquake magnitudes
1<m<9.
