  A detailed analysis of the finite-size effects on the bulk critical behaviour
of the $d$-dimensional mean spherical model confined to a film geometry with
finite thickness $L$ is reported. Along the finite direction different kinds of
boundary conditions are applied: periodic $(p)$, antiperiodic $(a)$ and free
surfaces with Dirichlet $(D)$, Neumann $(N)$ and a combination of Neumann and
Dirichlet $(ND)$ on both surfaces. A systematic method for the evaluation of
the finite-size corrections to the free energy for the different types of
boundary conditions is proposed. The free energy density and the equation for
the spherical field are computed for arbitrary $d$. It is found, for $2<d<4$,
that the singular part of the free energy has the required finite-size scaling
form at the bulk critical temperature only for $(p)$ and $(a)$. For the
remaining boundary conditions the standard finite-size scaling hypothesis is
not valid. At $d=3$, the critical amplitude of the singular part of the free
energy (related to the so called Casimir amplitude) is estimated. We obtain
$\Delta^{(p)}=-2\zeta(3)/(5\pi)=-0.153051...$, $\Delta^{(a)}=0.274543...$ and
$\Delta^{(ND)}=0.01922...$, implying a fluctuation--induced attraction between
the surfaces for $(p)$ and repulsion in the other two cases. For $(D)$ and
$(N)$ we find a logarithmic dependence on $L$.
