This paper deals with the relationships of the maxima, minima and zeros of two random functions of known autocorrelations and cross-correlation, based on the work of Rice(4). Ratcliffe(3) and Briggs and Spencer(2) discuss a similar problem in connexion with a ‘Phillips Record’ of an experiment in ionospheric reflexion. In this experiment there are two highly correlated reflected signals, their maxima coming close together, and the record shows the time lags between a maximum on one signal and a maximum on the other. Briggs and Page (1) have made an experimental study of the distribution of the differences between the positions of the maxima of two highly correlated random functions, using EDSAC to construct the functions. In §§ 3–7 the frequency distributions of intervals between successive zeros and maxima, and of the lengths of intercepts by a horizontal line, are considered. This has applications to the study of the fading of long-wave radio signals, where the tune differences between successive maxima of the amplitude have been investigated.