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 f = in the foregoing analysis. The conditions 

 of convergency for //, = + 1 determine N a , N 3 , N^,... exactly as 

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

 fore, by (20), 



which is equivalent to N 1 = cc . This equation determines the 

 admissible values of/(=<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 /z.. 



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

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



f = #" sin cos . cos (<r + &> + e) (1), 



where a- 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 cr n, exactly, and therefore /=^. 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 formulae 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 1} for which a = n 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 

 ' luni-solar ' diurnal tide. 



232 



