Depth of maximum of air-shower profiles at the Pierre Auger Observatory. I. Measurements at energies above 1017.8eV

We report a study of the distributions of the depth of maximum, X-max, of extensive air-shower profiles with energies above 10(17.8) eV as observed with the fluorescence telescopes of the Pierre Auger Observatory. The analysis method for selecting a data sample with minimal sampling bias is described in detail as well as the experimental cross-checks and systematic uncertainties. Furthermore, we discuss the detector acceptance and the resolution of the X-max measurement and provide parametrizations thereof as a function of energy. The energy dependence of the mean and standard deviation of the X-max distributions are compared to air-shower simulations for different nuclear primaries and interpreted in terms of the mean and variance of the logarithmic mass distribution at the top of the atmosphere.