SUMMARY An analytic expression is derived for the average probability density function of the time interval between two consecutive events in a continuum of time, for the case in which the instantaneous rate of occurrence of events contains a term varying sinusoidally with the time in addition to a constant term. Expressions are obtained also for the average values of the mean and variance of the time interval between two consecutive events. An application of the theory to the analysis of phenomena for which the data consist solely of the times of occurrence of isolated events in a continuum of time is discussed briefly. The mathematical statistics presented in this paper originated in a study of cosmic ray extensive air showers. In particular, a search was made for the possible existence of a periodic variation in the experimentally observed occurrence rate of extensive air shower events at a fixed position on the earth's surface. Of special importance is the case in which the average rate of occurrence of events is roughly comparable with the anticipated frequency of the periodic modulation. The present theoretical investigation indicates how the amplitude and frequency of a periodic modulation can be estimated solely from a knowledge of the measured time intervals between successive events. Since the theory may be applied to the analysis of other phenomena, for which the data consist only of the times of occurrence of isolated events in a continuum of time, the arguments are presented without specific reference to the cosmic ray problem. The first objective of the theory is to determine the probability density function (P.D.F.) of the time interval between two consecutive events in a continuum of time, for the case in which the instantaneous rate of occurrence of events contains a term varying sinusoidally with the time in addition to a constant term. In the special case in which the instantaneous occurrence rate is constant, the P.D.F. of the time interval between two consecutive events may be derived at once from the usual form of the homogeneous Poisson distribution. The particular instantaneous occurrence rate considered here generates a non-homogeneous Poisson distribution, from which the average value of the P.D.F. of the time interval between two consecutive events may be derived. This average P.D.F. is then used to find the average values of the mean and variance of the time interval between two consecutive events. Finally, consideration is given to an application of the above theory to the statistical analysis of any phenomenon which manifests itself as the occurrence of discrete events in a continuum of time, and for which a periodic modulation of the occurrence rate is anticipated. The measured time intervals between successive events are used to calculate the observed average probability of obtaining a time interval between two consecutive events which lies