366 . ON TIDE-GENERATING FORCES. 



spheroid of the second order, of the zonal type, having its axis passing through 

 the disturbing body. 



C. Owing to the diurnal rotation, and also to the orbital motion of the 

 disturbing body, the position of the tidal spheroid relative to the earth is 

 continually changing, so that the level of the water at any particular place 

 will continually rise and fall. To analyse the character of these changes, let 

 6 be the co-latitude, and o&amp;gt; the longitude, measured eastward from some fixed 

 meridian, of any place P, and let A be the north-polar-distance, and a the 

 hour-angle west of the same meridian, of the disturbing body. We have, 



then, 



cos S- = cos A cos 6 + sin A sin 6 cos (a + o&amp;gt;) .................. (ix), 



and thence, by (vii), 



+ ^Hsin 2 A sin 26 cos (a + o&amp;gt;) 



+ pTsin 2 Asin 2 0cos2(a + a))-|-C ..................... (x). 



Each of these terms may be regarded as representing a partial tide, and 

 the results superposed. 



Thus, the first term is 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 = , or = 90 35 16 . The amount of the 

 tidal elevation in any particular latitude varies as cos 2 A J. In the case 

 of the moon the chief fluctuation in this quantity has a period of about a 

 fortnight ; we have here the origin of the lunar fortnightly or declina- 

 tional 3 tide. When the sun is the disturbing body, we have a solar semi 

 annual tide. It is to be noticed that the mean value of cos 2 A-^ with 

 respect to the time is not zero, so that the inclination of the orbit of the 

 disturbing body to the equator involves as a consequence a permanent change 

 of mean level. Of. Art. 180. 



The second term in (x) is a spherical harmonic of the type obtained by 

 putting ?i = 2, s = l in Art. 87 (6). The corresponding tidal spheroid has as 

 nodal lines the meridian which is distant 90 from that of the disturbing 

 body, and the equator. The disturbance of level is greatest in the meridian 

 of the disturbing body, at distances of 45 N. and S. of the equator. The 

 oscillation at any one place goes through its period with the hour-angle a, 

 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. 



The third term is a sectorial harmonic (n = 2, a = 2), 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 place goes through 

 its period with 2a, i. e. in half a (lunar or solar) day, and the amplitude varies 

 as sin 2 A, being greatest when the disturbing body is on the equator. We 

 have here the origin of the lunar and solar semi-diurnal tides. 



