266 The Tide-generating Forces 



Doodson (1922, p. 305) and Rauschelbach (1924) have given a full de- 

 velopment of the tidal potential of the kind indicated above, whereas the 

 earlier classical developments of Darwin (1883, p. 49) and Boergen (1884, 

 p. 305) have only taken into account the influence of the slow motion of 

 the lunar node by slow variations in the amplitude and phase of the cor- 

 responding terms. For the rest, the fundamental idea is the same; it emanated 

 from Kelvin (1872), who was the first to substitute fictitious celestial bodies 

 for individual terms of the tide potential. 



It is not possible to give here in all its details a harmonic analysis, (see 

 for example Bartels, 1936, p. 309; and, 1957, p. 734). We will only mention 

 briefly the final result. Each term of the series corresponds to the equation 

 (VIII. 14) and Q is the sum of all terms of this equation, which have the form: 



Partial tide = numerical coefficient x 



, .. ff . . . (cos) ' rt (VIII. 17) 



geodetic coefficient { . } (argument) . 



[sin] 



The geodetic coefficients are only dependent upon the geographical 

 latitude 0; the most important are those derived from Q % 



G = \mR{\ — 3 sin 2 9?) , 

 G x = imRs'm2(p , 

 G 3 = %mRcos 2 q> . 



The argument or the phase {at — x) is an aggregate composed of the above- 

 mentioned six variables; they determine the frequency of the term and, hence, 

 its period. Thus, for instance, r represents lunar time, (r+s) sideral time, 

 (r + s—h) solar time, 2t the semi-diurnal lunar wave, etc. Abbreviating, the 

 partial tides are designated according to their argument, and this is given 

 symbolically by the "argument number". The argument (2r — 3s + 4h J r p — 

 — 2N+2p s ) is given by the argument number (229-637): the first figure gives 

 the factor of r, whereas the following figures are the factors of the other 

 variables increased by 5. The most important are the first 3 figures, inasmuch 

 as they give positions which repeat themselves within a year. The last figure 

 is generally left out. The first figure also gives the main distribution of the 

 partial tides: long periodical, 1 diurnal, 2 semi-diurnal, 3 the third diurnal 

 partial tides. 



The argument in (VIII. 17) is composed of two parts: the variable part t 

 (frequency x time) and the phase, which is the non-variable part of the 

 argument, which is given at the origin of time, usually the 1st of January 

 of a year at 00 00 h. 



The complete list of all components established by Doodson lists approx- 

 imately 390, of which about 100 are long periodic, 160 diurnal, 115 semi- 



