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 10arcmin 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 3arcmin 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 $441 < ell < 785$.
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