The parabolic-layer transmission equation of Appleton and Beynon has been solved for a wide range of its main parameters. A method of applying the equation to any long-distance circuit, in which the successive hop lengths of each transmitted ray are adjusted to accord with ionospheric variations along the path, is described. The only data required are directly obtainable from published ionospheric predictions. The results are displayed on mode plots, which comprise (a) a mode-angle plot, showing the angles of elevation of all rays and the modes in which they are propagated to all points along a great circle of any extent, and (b) a mode-delay plot, showing the time of travel of all rays instead of their angles of elevation. The effect on the mode plots of diurnal and longer-term changes in the ionosphere is discussed. Two long paths are examined experimentally and it is shown that the number of signal components observed, their relative time-delays, their angles of elevation and their response to varying ionospheric conditions, all agree well with those determined theoretically from the mode plots, provided that the effect of the tropical-Es layer is included. Mode-angle plots may also be used without modification to find both the angular spectrum of back-scatter patterns and the most probable paths taken by atmospheric noise energy arriving at a receiver. Mode-delay plots also provide direct backscatter patterns. The information on the plots may be used to improve aerial design and also to choose an optimum signalling frequency such that the number of active modes may be reduced, with a possible improvement in delay distortion of the signal. The choice of an optimum signal frequency is made possible by means of a mode-time plot, which shows the diurnal changes to be expected in propagation characteristics as determined by predicted changes in the ionosphere over the path.