Corrections to the classical tidal free-surface configuration are obtained for two cases of combined rotation and orbital translation. Lord Kelvin's calculation for rotation parallel to and commensurable with orbital revolution is presented in terms of an appropriate acceleration potential. Extension to fast rotation and inclined axes of rotation and orbital motion, based on Chasles' Theorem and more closely approximating the case of the Earth, follows in the same manner. In both cases the corrections to diurnal and to semi-diurnal tides vanish when the orbital plane of the disturbing body is contained within the Earth's equatorial plane, but the corrections are large for inclination values corresponding to actual Moon and Sun. Results in sharp contrast with the older theory in respect to amplitudes and phases are generally in better qualitative agreement with observations. The corrected theory should provide a more acceptable basis for the dynamical theory of ocean tides and also prove serviceable in allied studies of Earth physics, astronomy, and cosmogony.The dominant correction term in the fast-rotation tidal model is a spherical harmonic of the first degree, for which no counterpart exists in the standard tide theory. As this is of the same form precisely as the recently observed offset core of the Moon, tidal mechanics affords a new hypothesis to explain the latter phenomenon.