  We propose a procedure for the determination of the time-dependent velocity
and pressure fields of an unbounded incompressible viscous fluid in an external
force field induced by an arbitrary number of spheres moving and rotating in it
as well as the forces and torques exerted by the fluid on the particles. Within
the completely linearized scheme, we express the velocity and pressure fields
of the fluid in terms of induced surface force densities and derive the
explicit form for all quantities contained in these relations not imposing any
additional restrictions on the size of particles, distances between them, and
the frequency range. We show the incorrectness of similar results obtained
earlier by several authors because these results are expressed in terms of
nonexistent inverse tensors. We explain the reasons leading to this and propose
a procedure for the elimination of divergent quantities. In the stationary
case, using the proposed procedure, we obtained the translational, rotational,
and coupled friction and mobility tensors for a system containing an arbitrary
number of spheres up to, respectively, the second, forth, and third orders in
the dimensionless parameter equal to the ratio of a typical radius of a sphere
to a typical distance between two spheres. In various particular cases, the
results obtained in the present paper agree with the well-known results derived
by other methods.
