  We study the canonical solution of a family of classical $n-vector$ spin
models on a generic $d$-dimensional lattice; the couplings between two spins
decay as the inverse of their distance raised to the power $\alpha$, with
$\alpha<d$. The control of the thermodynamic limit requires the introduction of
a rescaling factor in the potential energy, which makes the model extensive but
not additive. A detailed analysis of the asymptotic spectral properties of the
matrix of couplings was necessary to justify the saddle point method applied to
the integration of functions depending on a diverging number of variables. The
properties of a class of functions related to the modified Bessel functions had
to be investigated. For given $n$, and for any $\alpha$, $d$ and lattice
geometry, the solution is equivalent to that of the $\alpha=0$ model, where the
dimensionality $d$ and the geometry of the lattice are irrelevant.
