302 The Harmonic Analysis of Tidal Observations 



numbers s and p nearest to this figure which satisfy (X.6) is obtained when 

 p = 738 and s = 713. This means that pT = 369 days, is the most favour- 

 able length of time to be selected for the harmonic analysis of the M 2 tide. 

 The periods of the tide being incommensurable, it is possible to analyse to 

 the fullest extent only one tide at a time, by selecting the proper interval, 

 and the influence of the other tides will become smaller, the greater the 

 interval. The time of 369 days proves to be the most favourable interval for 

 a great number of component tides with a short period, so that this interval 

 has generally been selected for the reduction of the tides. In case the obser- 

 vations extend over a shorter period of time, the procedure has to be changed 

 somewhat; however, this depends essentially upon the form in which the 

 material is available. 



The numerical computation of the average variation in sea level caused 

 by a partial tide, requires the division of the whole tidal curve into equal 

 intervals of one period (component days), whose lengths are the periods of 

 the various diurnal components or twice the periods of the semi-diurnals. 

 Such component days are divided into twenty-four equal parts called com- 

 ponent hours. The tidal curve is read at the component hours. Then follows 

 the averaging of all hourly values, in order to obtain the average variation 

 during a component day. By repeating this procedure for other periods, it is 

 possible to separate all partial tides from each other. However, such an analysis 

 would require extremely extensive, laborious and time-consuming compu- 

 tations and only the observations made with a self-recording tide gauge could 

 be used, with the exclusion of observations consisting in hourly readings of 

 the sea level. Roberts and Darwin have therefore given the following important 

 simplification of the procedure. 



The tidal curve is tabulated in mean solar hours, so that one obtains 

 24 sums of the water height which will serve all components. Then one distri- 

 butes the solar hourly heights among the component hours as nearly as possi- 

 ble. The speed or periods of the components determine where the various 

 component hours fall upon the solar hours. If s x is the hourly speed of the 

 mean sun (= 15°) or diurnal solar component, c x the speed for any other 

 diurnal component, then cjs t = cj\5 represents the portion of any com- 

 ponent hour corresponding to a solar hour. A half component hour will 

 be lost or gained accordingly as c is less or greater than s, when \s l j{s l ^c l ) 

 = 15/(30 — ^2c x ) solar hours shall have elapsed from the beginning. At sub- 

 sequent regular intervals of sJis^ c x ) = 15/(15^-'C 1 ) solar hours, a whole 

 component hour will be lost or gained, that is, the difference between com- 

 ponent and solar hours will increase one at such time. If c x < s x as is usually 

 the case, two adjacent solar hours at one of these times fall upon the same 

 component hour, i.e. within a half component hour of the time aimed at; 

 but if e x > s x a component hour will be skipped because no solar hour occurs 

 within a half component hour of it. If the maximum divergence allowed be 



