The Harmonic Analysis of Tidal Observations 313 



From (X. 11) we obtain for the time of the high water 



l5^ + 2-2 arCta %+^cos2 y - J- (X ' 13) 



The third term in the brackets is a semi-monthly inequality. At the time of 

 full and new moon a — a' = xp = and the 



time of high water =^o « — arc tan 



30°[ *"" — A+A'cos(x-x')\' (X,14) 



This is the approximate true local time of the high water at full and new 

 moon and is called the "vulgar establishment". To the contrary, 



is the "corrected" establishment or the mean high water lunitidal interval. 



Determining the vulgar establishment is very simple and, consequently, 

 it is known for many harbours. On the contrary, the computation of the 

 corrected establishment requires continuous observations of the sea level 

 (determination of the ^-number of M 2 ). The corrected establishment still 

 deviates from individual values of the lunitidal interval by considerable 

 amounts. Greater accuracy is obtained if, for each day of the lunar period, 

 one computes the specific lunitidal interval according to (X. 13) or derives 

 it statistically from lengthy observations. 



The expression for C in (X. 11) gives at spring time (time of occurrence 

 of the spring tide) 



C = A+A', 2y' = 0; from which follows that spring time = day 



before full and new moon— age of the tide. (X. 16) 



If Z is the height of the mean sea level above chart datum, the 



height of the high water = Zq+C — approximately 



Z + (A+A'cos2y>'). 



height of the low water = Z —C = approximately 



Z —(A J r A 'cos 2tp'). 



(X.17) 



It can be seen from equations (X. 11) to (X. 17) that if the lunitidal interval 

 and the true solar time of the transit of the moon through the meridian for 

 one day are known, one has all the factors necessary to determine the semi- 

 diurnal tide, insofar as they are dependent on the simple orbital motion 

 of the disturbing body in the equatorial plane. However in order that the 

 basic values computed therefrom can be compared with the actual ones, 

 they must be corrected, for the influence of the declination and the parallax 

 of moon and sun and then also for the daily inequality; these corrections 

 themselves vary with the declination of the disturbing bodies. For this reason, 

 this procedure can only be applied, where the diurnal inequality is small 



