ESDU 83021 presents analytical methods which allow, by a consideration of kinematics only, a design or analysis of an epicyclic gear train to be accomplished. The single-carrier gear train, the simplest form of epicyclic, is considered to consist of two co-axial gears, one planet carrier and one or more planet gears. The thirty four unique practical cases for this type of epicyclic are treated separately and information on each is presented in a separate table. For each of the six possible cases (depending on which element is fixed and which is used for input or output) for each configuration the table gives graphs of the achievable gear ratios and efficiencies, formulae for the planet bearing speeds and a schematic representation of the epicyclic train. An epicyclic gear train consisting of more than one planet carrier or more than two co-axial gears, or both, can be considered to be a series of single-carrier gear trains connected together and consequently can be analysed using combinations of the single-carrier tables.