  We reanalyze the deep Chandra observations of the M87 jet, first examined by
Wilson & Yang (2002). By employing an analysis chain that includes image
deconvolution, knots HST-1 and I are fully separated from adjacent emission. We
find slight but significant variations in the spectral shape, with values of
$\alpha_x$ ranging from $\sim 1.2-1.6$. We use VLA radio observations, as well
as HST imaging and polarimetry data, to examine the jet's broad-band spectrum
and inquire as to the nature of particle acceleration in the jet. As shown in
previous papers, a simple continuous injection model for synchrotron-emitting
knots, in which both the filling factor, $f_{acc}$, of regions within which
particles are accelerated and the energy spectrum of the injected particles are
constant, cannot account for the X-ray flux or spectrum. Instead, we propose
that $f_{acc}$ is a function of position and energy and find that in the inner
jet, $f_{acc} \propto E_\gamma^{-0.4 \pm 0.2} \propto E_e^{-0.2 \pm 0.1}$, and
in knots A and B, $f_{acc} \propto E_\gamma^{-0.7 \pm 0.2} \propto E_e^{-0.35
\pm 0.1}$, where $E_\gamma$ is the emitted photon energy and and $E_e$ is the
emitting electron energy. In this model, the index $p$ of the injected electron
energy spectrum ($n(E_{e}) \propto E_{e}^{-p}$) is $p=2.2$ at all locations in
the jet, as predicted by models of cosmic ray acceleration by ultrarelativistic
shocks. There is a strong correlation between the peaks of X-ray emission and
minima of optical percentage polarization, i.e., regions where the jet magnetic
field is not ordered. We suggest that the X-ray peaks coincide with shock waves
which accelerate the X-ray emitting electrons and cause changes in the
direction of the magnetic field; the polarization is thus small because of beam
averaging.
