  We extend the analysis of the MAXIMA-1 cosmic microwave background (CMB) data
to smaller angular scales. MAXIMA, a bolometric balloon experiment, mapped a
124 deg$^2$ region of the sky with 10\arcmin resolution at frequencies of 150,
240 and 410 GHz during its first flight. The original analysis, which covered
the multipole range $36 \leq \ell \leq 785$, is extended to $\ell = 1235$ using
data from three 150 GHz photometers in the fully cross-linked central 60
deg$^2$ of the map. The main improvement over the original analysis is the use
of 3\arcmin square pixels in the calculation of the map. The new analysis is
consistent with the original for $\ell < 785$. For $\ell > 785$, where
inflationary models predict a third acoustic peak, the new analysis shows power
with an amplitude of $56 \pm 7$ \microk at $\ell \simeq 850$ in excess to the
average power of $42 \pm 3$ \microk in the range $441 < \ell < 785$.
