  The existence of the thermodynamic limit in spin systems with short- and
long-range interactions is established. We consider the infinite-volume limit
with a fixed shape of the system. The variational expressions of the entropy
density and the free energy density are obtained, which explicitly depend on
the shape of the system. This shape dependence of thermodynamic functions
implies the nonadditivity, which is one of the most important characteristics
of long-range interacting systems.
