The tethered satellite system (TSS) operating in an elliptic orbit may have a periodic motion. The configuration and stability of the limit cycle depend on the control algorithm and the orbital eccentricity. The size of the limit cycle increases as the orbital eccentricity increases. At a critical value of the eccentricity and thereafter, the limit cycle becomes unstable. The numerical method for computing the periodic motion, the analysis of local stability, and the domain of attraction given in this note are very suitable for investigating system dynamics and control of TSS.