  The generalized Gibbs-Duhem equation is obtained for systems with long-range
interactions in $d$ spatial dimensions. We consider that particles in the
system interact through a slowly decaying pair potential of the form $1/r^\nu$
with $0\leq\nu\leq d$. The local equation of state is obtained by computing the
local entropy per particle and using the condition of local thermodynamic
equilibrium. This local equation of state turns out to be that of an ideal gas.
Integrating the relation satisfied by local thermodynamic variables over the
volume, the equation involving global magnitudes is derived. Thus, the Euler
relation is found and we show that it is modified by the addition of a term
proportional to the total potential energy. This term is responsible for the
modification of the Gibbs-Duhem equation. We also point out a close
relationship between the thermodynamics of long-range interacting systems and
the thermodynamics of small systems introduced by Hill.
