  We present 1998 HST observations of M87 which yield the first single-epoch
optical and radio-optical spectral index images of the jet at $0.15''$
resolution. We find $<\alpha_{ro} > \approx 0.67$, comparable to previous
measurements, and $<\alpha_o > \approx 0.9$ ($F_\nu \propto \nu^{-\alpha}$),
slightly flatter than previous workers. Reasons for this discrepancy are
discussed. These observations reveal a large variety of spectral slopes. Bright
knots exhibit flatter spectra than interknot regions. The flattest spectra
($\alpha_o \sim 0.5-0.6$; comparable to or flatter than $\alpha_{ro}$) are
found in two inner jet knots (D-East and HST-1) which contain the fastest
superluminal components. In knots A, B and C, $\alpha_o$ and $\alpha_{ro}$ are
essentially anti-correlated. Near the flux maxima of knots HST-1 and F, changes
in $\alpha_{ro}$ lag changes in $\alpha_o$, but in knots D and E, the opposite
relationship is observed. This is further evidence that radio and optical
emissions in the M87 jet come from substantially different physical regions.
The delays observed in the inner jet are consistent with localized particle
acceleration, with $t_{acc} << t_{cool}$ for optically emitting electrons in
knots HST-1 and F, and $t_{acc} \sim t_{cool}$ for optically emitting electrons
in knots D and E. Synchrotron models yield $\nu_B \gsim 10^{16}$ Hz for knots
D, A and B, and somewhat lower values, $\nu_B \sim 10^{15}- 10^{16}$ Hz, in
other regions. If X-ray emissions from knots A, B and D are co-spatial with
optical and radio emission, we can strongly rule out the ``continuous
injection'' model. Because of the short lifetimes of X-ray synchrotron emitting
particles, the X-ray emission likely fills volumes much smaller than the
optical emission regions.
