78 RECORD AND REDUCTION OF THE TIDES. 



of the inequality corresponding to an increasing (irrespective of sign) declination 

 of the moon, but the curve appears double-crested about the time of maximum 

 declination, there being a sudden diminution in the inequality, preceded and fol- 

 lowed by high values; about the time of the moon's crossing the equator the 

 inequality is very irregular. 



On Plate IV, the actual separation of the semi-diurnal and the diurnal wave has 

 been effected graphically, for which purpose a part of the best observations was 

 selected; these observations extend over the period from Oct. 30 to Nov. 22, 1853. 

 The process of decomposition in use in the U. S. Coast Survey was at first an 

 analytical one, by computing sine curves ; since 1855, however, a graphical process, 

 equivalent thereto, was substituted ; this latter method, as introduced by assistant 

 L. F. Pourtales, may be briefly explained as follows : After the observations are 

 plotted and a tracing is taken, the traced curves are shifted in epoch 12 (lunar) 

 hours forward, when a mean curve is pricked off between the observed and traced 

 curves; this process is repeated after the tracing paper has been shifted 12 hours 

 backward; the average or mean pricked curve thus obtained represents the semi- 

 diurnal wave. On an axis parallel with that on which the time is counted, the 

 differences between the originally observed and the constructed semi-diurnal wave 

 were laid off; this constitutes the diurnal curve. In the case in hand I have 

 simplified the process of separation by blackening the under surface of the tracing 

 paper with a lead pencil, and running in with a free hand ; the intermediate curves 

 by the pressure of a style, an average of the two traces thus left on the lower paper, 

 gave the semi-diurnal wave in quite an expeditious manner. On the diagram, the 

 diurnal curve with its epoch of high water nearly coinciding with that of the semi- 

 diurnal wave, appears plainly with its variation in size depending on the moon's 

 declination. 



Investigation of the Form of the Tide Wave. — The shape of the tide wave has 

 been ascertained in the manner described in art. (479) Tides and Waves, and 

 depends on the hourly observations of 60 tides, 30 during spring tides and an 

 equal number during neap tides, that is, the observed heights on the day of the 

 syzygies and quadratures and on the first and second day after, were tabulated, 

 forming ten groups of three columns each, from low water to low water. The 

 columns of an equal number of hours (they vary from Hi hours to 11 hours) were 

 united in a mean. In order to combine these it Avas assumed that the interval from 

 the observed low water to the next following low water corresponds to 360° of 

 phase, and the time of every intermediate observation was converted into phase by 

 that proportion. In order to render the observed heights comparable, the range 

 from high to low water in every half tide (the reading of low water for phase 

 generally not being identical with the reading of the succeeding low water or phase 

 360°) was supposed to correspond to 2.00, and the elevation above the low water 

 was converted into number by that proportion, thus furnishing a series of ordinates 

 for equidistant abscissa?. The means of all the phases and corresponding converted 

 depressions within every 30th degree of phase were then taken with proper regard 

 to the weights, depending on the number of columns, of equal hours, united at the 

 commencement of the reduction. By observation of the progress of the numbers, 



