368 ON TIDE-GENERATING FORCES. 



expansion, taking account of the variations of A and a, and of the distance 

 D of the disturbing body, (which enters into the value of #), is a somewhat 

 complicated problem of Physical Astronomy, into which we do not enter here*. 



Disregarding the constant (7, which disappears in the dynamical equations 

 (1) of Art. 207, the constancy of volume being now secured by the equation of 

 continuity (2), it is easily seen that the terms in question will be of three 

 distinct types. 



First, we have the tides of long period, for which 



=#'(cos 2 0-i).cos(o-* + ..................... (xiii). 



The most important tides of this class are the * lunar fortnightly ' for which, 

 in degrees per mean solar hour, o- = l'098, and the 'solar semi-annual' for 

 which <r = 0-082. 



Secondly, we have the diurnal tides, for which 



where a differs but little from the angular velocity n of the earth's rotation. 

 These include the 'lunar diurnal' [&= 13 -943], the 'solar diurnal' [a = 14 -959], 

 and the 'luni-solar diurnal' [0- = ft = 15 -041]. 



Lastly, we have the semi-diurnal tides, for which 



f=#"'siii 2 0.cos(<r* + 2a> + e) ..................... (xv)f, 



where or differs but little from 2w. These include the ' lunar semi-diurnal ' 

 [0- = 28 -984], the 'solar semi-diurnal' [o-=30], and the 'luni-solar semi- 

 diurnal ' [o- = 2w = 30 -082]. 



For a complete enumeration of the more important partial tides, and for 

 the values of the coefficients H', H" , H'" in the several cases, we must refer 

 to the papers by Lord Kelvin and Prof. G. H. Darwin, already cited. In the 

 Harmonic Analysis of Tidal Observations, which is the special object of these 

 investigations, the only result of dynamical theory which is made use of is the 

 general principle that the tidal elevation at any place must be equal to the 

 sum of a series of simple-harmonic functions of the time, whose periods are 

 the same as those of the several terms in the development of the disturbing 

 potential, and are therefore known a priori. The; amplitudes and phases of 

 the various partial tides, for any particular port, are then determined by 

 comparison with tidal observations extending over a sufficiently long period J. 



* Reference may be made to Laplace, Mecanique Celeste, Livre 13 me , Art. 2 ; to 

 the investigations of Lord Kelvin and Prof. G. H.JDarwin in the Brit. Ass. Reports 

 for 1868, 1872, 1876, 1883, 1885 ; and to the Art. on " Tides," by the latter author, 

 in the Encyc. Britann. (9th ed.). 



t It is evident that over a small area, near the poles, which may be treated as 

 sensibly plane, the formulas (xiv) and (xv) make 



f xrcos(<r + a> + e), and f <x r 2 cos (<rt + 2w + e), 



respectively, where r, w are plane polar coordinates. These forms have been used 

 by anticipation in Arts. 188, 204. 



J It is of interest to note, in connection with Art. 184, that the tide-gauges, 

 being situated in relatively shallow water, are sensibly affected by certain tides of 

 the second order, which therefore have to be taken account of in the general 

 scheme of Harmonic Analysis. 



