The fundamental relations (connecting the torques, the velocities and the power flows) for a general differential mechanism of imperfect efficiency are newly derived from first principles. The power flow relations, which have not previously been published, are shown to be a useful tool for the analysis of complex chains of differential gears and they also permit the immediate derivation of the efficiency of all the inversions of an epicyclic gear. The range of applicability of the power relations is investigated and clearly defined, and results obtained by their use are tabulated.