This paper presents a systematic method of developing all the possible configurations for the assumed sinusoidal input to the sampler, as the lag varies between 0°and 360°and for the output of the zero-order hold following a relay with dead zone. Any period of oscillation characterized by a different number of positive and negative relay corrections can be considered. The equivalence of the critical regions for different periods of oscillation is pointed out. The critical points on the waveform at which either a relay correction or an entry into the dead zone can occur are precisely determined. Further the critical instants of sampling during a period of oscillation are also clearly established with the result that the development of the critical regions is made much easier. The meeting of the maxima and minima of two different critical regions is indicated, thus aiding in the correct development of the critical regions. The size of the area of different critical regions for the same period of oscillation and the total number of critical regions obtainable for any period of oscillation are discussed. The critical regions are developed for the periods of oscillation of 9T through 20T based on the above considerations. These critical regions together with the recently reported result in respect of an odd period of oscillation are made use of to identify all the possible modes that can be sustained in a second-order system with transportation lag and a third-order system. Thus the utility of the discrete describing function is considerably enhanced.