We discuss the cosmological implications of the new constraints on the power spectrum of the Cosmic Microwave Background Anisotropy derived from a new high resolution analysis of the MAXIMA-1 measurement (Lee et al. 2001). The power spectrum shows excess power at $ell sim 860$ over the average level of power at $411 leell le 785.$ This excess is statistically significant on the 95% confidence level. Such a feature is consistent with the presence of a third acoustic peak, which is a generic prediction of inflation-based models. The height and the position of the excess power match the predictions of a family of inflationary models with cosmological parameters that are fixed to fit the CMB data previously provided by BOOMERANG-LDB and MAXIMA-1 experiments (e.g., Jaffe et al.2001). Our results, therefore, lend support for inflationary models and more generally for the dominance of coherent perturbations in the structure formation of the Universe. At the same time, they seem to disfavor a large variety of the non-standard (but still inflation-based) models that have been proposed to improve the quality of fits to the CMB data and consistency with other cosmological observables. Within standard inflationary models, our results combined with the COBE-DMR data give best fit values and 95% confidence limits for the baryon density, $Omega_b h^2simeq 0.033{pm 0.013}$, and the total density, $Omega=0.9{+0.18atop -0.16}$. The primordial spectrum slope ($n_s$) and the optical depth to the last scattering surface ($tau_c$) are found to be degenerate and to obey the relation $n_s simeq 0.46 tau_c + (0.99 pm 0.14)$, for $tau_c le 0.5$ (all 95% c.l.).
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