  The dielectric sphere has been an important test case for understanding and
calculating the vacuum force of a dielectric body onto itself. Here we develop
a method for computing this force in homogeneous spheres of arbitrary
dielectric properties embedded in arbitrary homogeneous backgrounds, assuming
only that both materials are isotropic and dispersionless. Our results agree
with known special cases; most notably we reproduce the prediction of Boyer and
Schwinger et al. of a repulsive Casimir force of a perfectly reflecting shell.
Our results disagree with the literature in the dilute limit. We argue that
Casimir forces can not be regarded as due to pair-wise Casimir-Polder
interactions, but rather due to reflections of virtual electromagnetic waves.
