  The Casimir effect giving rise to an attractive force between the closely
spaced two concentric spheres that confine the massless scalar field is
calculated by using a direct mode summation with contour integration in the
complex plane of eigenfrequencies. We devoleped a new approach appropriate for
the calculation of the Casimir energy for spherical boundary conditions. The
Casimir energy for a massless scalar field between the closely spaced two
concentric spheres coincides with the Casimir energy of the parallel plates for
a massless scalar field in the limit when the dimensionless parameter {\eta},
({\eta}=((a-b)/(\surd(ab))) where a (b) is inner (outer) radius of sphere),
goes to zero. The efficiency of new approach is demonstrated by calculation of
the Casimir energy for a massless scalar field between the closely spaced two
concentric half spheres.
  PACS number(s): 03.70.+k, 12.20.DS, 11.10.Gh
