64 U. S. COAST AND GEODETIC SURVEY. 



division be numbered consecutively, beginning with zero at the initial 

 instant of each original division. Because of the exact reproduction 

 of the period of component A in each division, it is evident that the 

 instants with hke numbers in the different divisions will correspond 

 to the same phase of component A but to unlike phases of each of the 

 other components. At each such instant the height of the observed 

 tide will equal the height of component A for the phase corresponding 

 to that instant, plus the heights of all of the other components, 

 together with any variations due to meteorological causes. 



Assuming, for convenience, mean sea level as a datum and adding 

 the heights of the tide corresponding to the instants of like numbers in 

 each division it is evident that there will be included in the sum a cer- 

 tain number of equal heights corresponding to a particular phase of 

 component A, together with the unequal heights corresponding to 

 various phases of the other components and also the meteorological 

 fluctuations. In a series of sufficient length the sum of the unequal 

 heights of each of the disturbing components and of the meteorological 

 fluctuations will become zero, since the positive heights will be offset 

 by the negative heights, while the equal heights corresponding to the 

 particular phase of component A will accumulate as the summing 

 proceeds. The average height of the tide obtained from such a sum 

 when a limited series of observations has been used will be equal to the 

 height of component A corresponding to a particular phase, plus a 

 small residual due to the imperfect elimination of the disturbing com- 

 ponents and meteorological effects. If the average height correspond- 

 ing to each subdividing instant of the division is obtained and plotted 

 as an ordinate with the time subdivisions as abscissae, a curve may be 

 drawn which will approximately represent the component A through- 

 out its entire period. If the heights of the original tabulation are 

 referred to any arbitrary datum having a definite relation to mean sea 

 level, the summation will be equally effective in eliminating the dis- 

 turbing elements, and the resulting heights for component A will be 

 referred to that datum. 



If, instead of making each division of the series equal to a single 

 period of the component sought, we let it include any exact integral 

 number of periods, the same principles will apply for the elimination 

 of the other components. In this case the resulting average heights 

 when plotted will represent the corresponding number of periods of 

 the component sought. As a matter of convenience in the harmonic 

 analysis of the short period components it is customary to include in 

 each division such multiples of the component period as will most 

 nearly conform to the solar day. Such a division is a component day. 

 If the component is diurnal, its component da}" will include exactly 

 one period; if semidiurnal, exactly two periods; if terdiurnal, exactly 

 three periods, etc. Each component day is subdivided into 24 equal 

 parts, each part being designated as a component hour. The initial 

 instants of the component hours of each component day are num- 

 bered consecutively from to 23. For the long-period components 

 the original divisions are taken as the component month or the com- 

 ponent year, and these are subdivided into 24 equal parts. As the 

 long-period components require a different treatment than the short- 

 period components, they will be given special consideration in a later 

 section. 



