210-212] FREE OSCILLATIONS. 355 



211. It remains to notice how the free oscillations are deter 

 mined. In the case of symmetry with respect to the equator, we 

 have only to put H = in the foregoing analysis. The conditions 

 of convergency for yu, = + 1 determine N z , N 3 , JV 4 , ... exactly as 

 before; whilst equation (12) gives N 2 = 1 /3/ 2 /2 . 3, and there 

 fore, by (20), 



__ 



- ...... &amp;lt; 28 &amp;gt;&amp;lt; 



which is equivalent to ^ = x . This equation determines the 

 admissible values of / 1 = &amp;lt;r/2w). The constants in (9) are then 

 given by 



where A is arbitrary. 



The corresponding theory for the asymmetrical oscillations 

 may be left to the reader. The right-hand side of (8) must now 

 be replaced by an even function of yu,. 



212. In the next class of tidal motions (Laplace s Oscillations 

 of the Second Species ) which we shall consider, we have 



fjr mooe0.&amp;lt;xw(&amp;lt;r* + + e) ............... (1), 



where &amp;lt;r differs not very greatly from n. This includes the lunar 

 and solar diurnal tides. 



In the case of a disturbing body whose proper motion could be 

 neglected, we should have o- = n, exactly, and therefore /= J. In 

 the case of the moon, the orbital motion is so rapid that the actual 

 period of the principal lunar diurnal tide is very appreciably 

 longer than a sidereal day*; but the supposition that/=J sim 

 plifies the formula so materially that we adopt it in the following 



* It is to be remarked, however, that there is an important term in the harmo 

 nic development of fi for which = 11 exactly, provided we neglect the changes in 

 the plane of the disturbing body s orbit. This period is the same for the sun as for 

 the moon, and the two partial tides thus produced combine into what is called the 

 1 luni-solar diurnal tide. 



232 



