264 The Tide-generating Forces 



this relation can be written 



cos# = cos^cos-J + sin^sinz]cos(/ + A-180°). (VIII.12) 



If we introduce this expression in (VIII. 4), we obtain, if \mR = A 



Q = ^(cos 2 Zl-i)(3cos 2 /-l)-^sin2<5sin20cos(/ + A) + 



+y4cos 2 (5cos 2 0cos2(? + A) . (VIII. 13) 



Each of these terms can be considered as representing a partial tide (Lamb, 

 1932, p. 359). 



(1) The first term does not contain the hour angle. As the declination varies only very slowly, 

 it can be regarded for a short time as constant. Tn regard to /, i.e. the geographical latitude <p, it 

 represents (see P 2 (6) in VIII. 8) a zonal harmonic of the second order and gives a tidal spheroid 

 symmetrical with respect to the earth's axis having as nodal lines the parallels for which cos 2 ^ = i 

 or 4>=±35°\6'. The amount of the tidal elevation in any particular latitude varies as 

 cos*A-l = £(cos2J + i). 



The main declinational inequality has, in case of the moon, a period of one-half of the period 

 of the variation of the declination, i.e. one half tropic month. We have here the origin of the lunar 

 fortnightly or 'declinationar tide. When the sun is the disturbing body, we have a solar semi-an- 

 nual tide. 



(2) The second term is a spherical harmonic. The corresponding tidal spheroid has as nodal 

 lines the meridian of which is distant 90° from that of the disturbing body, and the equator (</> = 0). 

 The disturbance of level is greatest in the meridian of the disturbing body, at distances of 45° N. 

 and S. of the equator (Tesseral harmonic P' 2 ). The oscillation at any one place goes through its 

 period with the hour angle, i.e. in a lunar or solar day. The amplitude is, however, not constant, 

 but varies slowly with A, changing sign when the disturbing body crosses the equator. This term 

 accounts for the lunar and solar 'diurnal' tides. 



(3) The third term is a sectorial harmonic i 32 and gives a tidal spheroid having as nodal lines 

 the meridians which are distant 45 E. and W from that of the disturbing body. The oscillation at 

 any one place goes through its period with 2x, i.e. in half a (lunar or solar) day, and the amplitude 

 varies as cos 2 d, being greatest when the disturbing body is on the equator. We have here the origin 

 of the lunar and solar 'semi-diurnal' tides. 



The expression (VIII. 13) shows the essential properties of the tide potential 

 which varies with time, but it is not entirely satisfactory. Both the decli- 

 nation <3 and the amplitude A are variable with time, because also the fluctuations 

 in the distance between the earth and the disturbing body enter in the value m. 

 A complete harmonic analysis of the tide potential requires Q to be expanded 

 in a series of simple cosine or sine-functions, with constant amplitudes and 

 constant periods. The actual periods (e.g. lunar day and solar day) are not 

 integer multiples of one fundamental period, but are incommensurable, and 

 such an expansion is most intricate. The derivation of each expansion requires 

 long and extensive computations. We will give here the fundamental idea. 



Let us assume that a simple term in Q has the form 



Q 1 = Ucos(at-x), (VIII. 14) 



