Analytical expressions are derived to show the geometric, thermodynamic, and aerodynamic relations among compressor, turbine, and exhaust nozzle for a gas-turbine engine. For a known compressor performance map, a matching method is described to show some of the design compromises that must be made when the components are to be combined into a turbine-propeller engine. A method of predicting engine performance for a range of operating conditions from known component maps is presented. An illustrative example of the matching method and the performance analysis is presented, showing some of the practical limitations of engine operation.