The Harmonic Analysis of Tidal Observations 301 



The partial tide has the index (amplitude H , phase « 0J period T = 2n/o ), 

 whereas the other tides are together under the summation sign; H cosx = B 

 and H n s'mx = C . 



If we multiply (X.2) once by cosr7 /, another time by smn^t, and if we 

 integrate these expressions from t = to t = pT {) , where p is a large number, 

 hence over a large number of periods of the partial tide <r , we obtain 



"Jo 



cos , B p V H 



r) . a tdt = pr ~ T ± 



sin ° C;2* ,a -^J *;-<* 



" = 1 



COS (7„ _ \ + cos 



. 2/wr — x„ . x, 



sin \<x ^ / — sin 



(X.3) 



These relations show that the computation of B and C„, which determine 

 the amplitude //„ and the phase « of the partial tide (t , become more accurate 

 when pT increases, i.e. the accuracy increases with the number of full periods 

 of the partial tide a which are used to compute averages. We then obtain 

 with a good approximation 



u cos 2 T cos *i4 rv a\ 



H ° sin "• = W« J n M a °""- (X ' 4) 







It is also possible to determine the smallest number of complete periods 

 which are necessary to eliminate a certain partial tide to have the expression 

 (X.4) become valid. The influence of other partial tides on a selected partial 

 tide which is to be obtained as clearly as possible, becomes important when 

 the frequencies approach each other. Thus, for instance, the S 2 tide will prove 

 to be particularly disturbing in computing the M 2 tide. However, this influ- 

 ence will vanish when in (X.3) the term below the summation sign for this 

 partial tide becomes zero. This happens for the disturbing a n component, 

 when 



— 2pjt approximates 2sn 



Consequently 



-. (X.5) 



P 



In which both 5 and p should be large integers. 



If we take, for instance, the S 2 tide, then a n = 15°, whereas for the M 2 tide 

 <j = 14-492051°; therefore, in this case, 



— = 103505 average solar hours. (X.6) 



If observations are available extending over a period of about one year, and 

 if this complete material is to be used, then pT is approximately 360 days, 

 and with T = \ day p is aproximately 720. The ratio of two whole 



