  We present an exact solution of the anisotropies of cosmic background
radiation (CBR) from a local collapse described by a spherical over-dense
region embedded in a flat universe, with the emphasis on the relationship
between the dipole $(\Delta {\sf T}/{\sf T})_d$ and the quadrupole $(\Delta
{\sf T}/{\sf T})_q$ anisotropy. This result has been used to examine the
kinematic quadrupole correction $(\Delta {\sf T}/{\sf T})_q=(\Delta {\sf
T}/{\sf T})_d^2/2$, which is usually applied to remove the contamination of the
quadrupole produced by local density inhomogeneities when finding the cosmic
amplitude of the quadrupole at the surface of last scattering. We have found
that the quadrupole of local collapse origin cannot always be approximately
described by the kinematic quadrupole. Our numerical result shows that the
difference between the kinematic and local quadrupoles depends on the size and
matter density in the peculiar field, and the position of the observer. For a
given dipole, the local quadrupole can be different from the kinematic
quadrupole by a factor as large as 3. Therefore, the kinematic quadrupole
correction remains an uncertain factor in the determination of the amplitude of
a cosmic quadrupole. Nevertheless, a preliminary analysis shows that this
uncertainty might not dramatically change the cosmological origin of the
COBE-DMR's quadrupole, unless a huge peculiar gravitational field is assumed.
