382 REPORT— 1892. 



This continuous line is the result of taking any two quantities which 

 gradually change, and may be plotted by other methods in other ways 

 such as those already mentioned. There is no limit to the number of 

 such lines or rows of points, except the very practical one of convenience. 

 Hence we might, instead of having the tide for one day, as shown in 

 fig. 9, have expressed graphically the tides for various different days in 

 succession, as in fig. 10 (which, together -with fig. 9 and other informa- 

 tion, has been kindly supplied by Mr. Anthony G. Lyster, assistant- 

 engineer, Mersey Docks and Harbour Board), which represents the suc- 

 cessive states of the tide from May 1 to May 12, 1892 ; but in the latter 

 case it becomes necessary to distinguish each particular curve in some 

 way, for instance as shown in the figure, by writing the corresponding 

 day upon each curve, just as it was necessary to do when a number of 

 distances were employed to represent isolated facts, as in figs. 4 and 5. 



The practical limit is placed by the number of curves which can be 

 recorded on a single diagram without producing confusion ; for instance, 

 suppose that instead of plotting the tidal results for twelve days, as in 

 the diagram given, for any date, it had been desired to give them for 

 every possible date — as, for instance, might be calculated by Lord Kelvin's 

 tide-predicter — it is quite clear that, even for one year, it would have been 

 quite impossible to do so without destroying the clearness of the diagram ; 

 and taking the time over a suSicient period would have merged the 

 curves into one another, and covered the surface of the paper. 



The truth is, there are now three variable quantities, the changes in 

 which it is desired to represent by the changing position of the point in 

 space, and this can no longer be done in a plane surface. The limit of 

 what can be represented by a true graphic method has been reached, and 

 models or some equivalent process of representation, such as shading, 

 equivalent to the delineation of the surface of a solid occupying three 

 dimensions of space, must be employed. A further illustration will make 

 this point clearer. 



Let fig. 11 represent another way of plotting similar tidal data, 

 the days of the month being now represented as the abscissae ; tiien for 

 a given spot (in the case taken, the dock-sill at Liverpool) and for 

 a given time (November 1888) we can trace for every day the time at 

 which the tide has reached any particular height at the dock-sill. Take, 

 for instance, the case of the tide when it is at datum level, which is shown 

 in the figure by dotted lines, and marked zero. This for the first day of 

 the month in question occurred at 1.30 a.m., on the 6th at 3.40 a.m., on 

 the 12th at 1.55 p.m., and so on ; but, moreover, as it occurred four times 

 in the twenty-four hours, these facts are recorded along the four different 

 lines, which are distinguished in the way mentioned. 



Now, if it be desired to trace the lines for each day at which any 

 other levels occur, it is only necessary to draw fresh lines. For instance, 

 let the case be taken at which the height is 4 feet below the sill ; this is 

 shown by a series of curves in dashed and dotted lines, thus — • — • — • • , 

 and marked —4. The times at which the tide is at the level of —4 on the 

 day abovementioned are: on the 1st, 2.45 A.M.; on the 5th, at 5 a.m. ; but 

 on the 12th the curves do not intersect the ordinate at all, showing that on 

 the twelfth day of the month — that is, during neap tides — the level of the 

 tide never falls 4 feet below the sill. A number of heights can be taken, 

 as shown in the diagram, which gives them for differences of level of 

 4 feet, and it is clear that if any smaller differences of height are required 



