  A detailed investigation of the scaling properties of the fully finite ${\cal
O}(n)$ systems with long-range interaction, decaying algebraically with the
interparticle distance $r$ like $r^{-d-\sigma}$, below their upper critical
dimension is presented. The computation of the scaling functions is done to one
loop order in the non-zero modes. The results are obtained in an expansion of
powers of $\sqrt\epsilon$, where $\epsilon=2\sigma-d$ up to ${\cal
O}(\epsilon^{3/2})$. The thermodynamic functions are found to be functions the
scaling variable $z=RL^{2-\eta-\epsilon/2}U^{-1/2}$, where $R$ and $U$ are the
coupling constants of the constructed effective theory, and $L$ is the linear
size of the system. Some simple universal results are obtained.
