312 The Harmonic Analysis of Tidal Observations 



hourly readings. Especially the luni-lidal intervals are taken into account, and 

 the procedure varies according to whether the different "establishments" are 

 used. It is, therefore, based principally upon the semi-diurnal components 

 and can only be used to advantage for harbours in which the diurnal inequality 

 is small. If this inequality is great, an additional number of corrections is 

 required to obtain usable results. The method fails completely where diurnal 

 tides prevail. The method is based on equation (IX. 2), where we have the 

 superposition of the semi-diurnal lunar tide with the semi-diurnal solar tide 

 for the equilibrium theory. In reality, each tide lags against the position of 

 the celestial body in the sky which is considered by introducing x numbers. 

 The equation (IX. 2) is, therefore, replaced by 



r\ = Acos[2(0-a)-x]+A'cos[2{6-a')-x'] . (X.9) 



The values with the prime are for the sun. d is the sidereal time, a and a' the 

 right ascensions of the moon and sun respectively, x = Q — a and x = Q — a 

 are the hour-angles of the moon and the sun and we have x = x'—y^ 

 if ip = a — a is the difference in hour angles of the moon and sun. We still 

 wish to mention that 2x = a M t and 2x' = a s t in which a M and o s are the 

 frequencies of the semi-diurnal lunar and solar tide respectively. t/15° is the 

 time expressed in lunar hours, t'/15° are solar hours = t hours. 



In this way (X.9) becomes 



rj =Acos[2(T'-y>)-x]+A'cos[2i;'-x']. (X.10) 



The two terms on the right can be contracted into a cos member with a some- 

 what variable amplitude and phase and we obtain 



in which 



rj = Ccos|2{(t'-v0-^+j8} 



C = [A 2 +A' 2 + 2AA'cos2y'] 1 i 2 , 



tan2fl = A+A-cos2y ' y =1'+-2- 



(X.ll) 



When the moon goes through the local meridian, the angular distance 

 of the sun is y) and the time t'/15° but the angular distance of the high water 

 caused by the sun and the moon is y>-hl(x— «')• * an d «' are constant, but y> 

 varies 0-5080°/h and 12192° or 0-81°/h day 



w = x'-x = \{gm-gs) = 15°- 14-492° = 0-508° 

 The variation of 2ip', which makes /S periodical, becomes 



2y' = (a s — Om) 

 if £ = 0041 («'— x) represents the age of the tide 



,f 7 [ '7 ] (X.12) 



14-77 days 



