The Harmonic Analysis of Tidal Observations 303 



assumed to be a half solar hour, then all solar hours are not represented by 

 component hours when c t < s; and when c x > s x a solar hour may oc- 

 casionally be taken to represent each of two consecutive component hours. 

 For each partial tide printed blanks can be used, in which is indicated at 

 certain spots by special marks where the hourly heights are to be entered. 

 These blanks do indicate the places where an hourly height has to be entered 

 in two successive columns, or where two successive hourly heights have to 

 be entered in the same column. The labour of writing still remains con- 

 siderable. Excellent forms for this computation can be found in Hann (1939, 

 p. 85) and Pollack (1926 or 28). Each variation in mean sea level given by 

 the mean values for each component hour is then developed into a series 

 of simple, double, etc., speed of the considered tide. We finally obtain, ac- 

 cording to the usual procedure of the Fourier Series 



rj = p^+p[ cos ( (Jl t-x[)+p:,cos(2o l t-x! 2 )+ ...+P^cos(n<r 1 t-x'„)+ ... (X.7) 



The hourly values of the spearate component hours of a partial tide obtained by the above- 

 mentioned method do not give exactly the sea level at the indicated full tidal hour, but the mean 

 sea level at all times during an interval of a half hour before and a half hour after the full tidal hour. 

 Therefore, the resulting mean value will be a trifle smaller (numerically) than the true ordinate. 

 To obtain the true value, we need to introduce an augmenting factor. If the correct presentation of 

 ij is given by 



>] =- P + PiCOs(f7 l t~y. 1 ) + P 2 cos(2a 1 t—y 2 ) + ...-rP n cos(,na 1 t — y. ll )+... (X.8) 



we obtain (X.7) by substituting each value in (X.8) by the average value for the interval from a half 

 hour before until a half hour after the full hour. As an hour is given by 7724, this means the 

 integration of (X.8) extends (/— T)/48 to (/+T)/48 and subsequent division by the entire interval 

 7/24. As we seek P„, we generally obtain 



, /CT/24 



p = p' 



sin(wr/24) 



so that the fraction to the right is the corrective factor which is to be applied to each term of equa- 

 tion (X.7) in order to obtain corrected augmented values. The phases remain unchanged [x = x'} 

 These augmenting factors are for 



/■• 



They are to be applied to all short period components, excepting the 5 -tides, where no augmenta- 

 tion is required, because the hourly values used were derived from the value at that particular full 

 solar hour. Therefore, in the analysis of the S- tide, each hourly height of the original tabulation 

 is used once and once only. 



If the term PiCos(a x t — xj represents in (X.7) the considered partial tide, 

 the following terms give the overtides in the same way as the upper harmonics 

 in the theory of accoustics; their frequencies are integer multiples of the 

 frequencies of the fundamental tide. This frequency agrees with that of the 

 tide potential. The occurrence of such overtides is due to the fact that, in 



