1. A chart of co-tidal and co-range lines for the North Sea was prepared at the Tidal Institute in the year 1923, and the methods then used have now been further developed and improved and applied to the English Channel, the Irish Sea, and their approaches. The methods used depend largely upon the known dynamical equations connecting the currents with the gradients of the elevations. If any assumption be made as to the values of the range of tide and the relative phases of the elevation and currents, then we can deduce from this information at one station, not only the directions of the co-tidal and co-range lines at that point, but also the degree of separation of the lines for any given unit of phase or range. This criterion, applied to speculative charts hitherto published, suffices at once either to verify or to condemn the charts. Again, without any assumptions at all, from the gradients at a number of stations on a line of small curvature, starting at a point at which the elevation is known, the elevations at all points along the line can be computed by simple methods of numerical integration.