  We review the derivation of the metric for a spinning body of any shape and
composition using linearized general relativity theory, and also obtain the
same metric using a transformation argument. The latter derivation makes it
clear that the linearized metric contains only the Eddington $\alpha$ and
$\gamma$ parameters, so no new parameter is involved in frame-dragging or
Lense-Thirring effects. We then calculate the precession of an orbiting
gyroscope in a general weak gravitational field, described by a Newtonian
potential (the gravito-electric field) and a vector potential (the
gravito-magnetic field). Next we make a multipole analysis of the potentials
and the precession equations, giving all of these in terms of the spherical
harmonics moments of the density distribution. The analysis is not limited to
an axially symmetric source, although the Earth, which is the main application,
is very nearly axisymmetric. Finally we analyze the precession in regard to the
Gravity Probe B (GP-B) experiment, and find that the effect of the Earth's
quadrupole moment (J_2) on the geodetic precession is large enough to be
measured by GP-B (a previously known result), but the effect on the
Lense-Thirring precession is somewhat beyond the expected GP-B accuracy.
