208-209] CASE OF SYMMETRY ABOUT AXIS. 349 



and the solar semi-annual tides, and, generally, all the tides of 

 long period. Their characteristic is symmetry about the polar axis. 



Putting s = in the formulae of the preceding Art. we have 



i<r df 



U ~ A> ( fa _ c2 ff\ JQ 



(2), 



<T COS 6 



V = 



, . , 1 d (hu sin 6) /ox 



and io-= --- ^-7, ^ ; - - .................. (3). 



a sm dd 



The equations (2) shew that the axes of the elliptic orbit of 

 any particle are in the ratio of / : cos 6. Since / is small, the 

 ellipses are very elongated, the greatest length being from E. to 

 W., except in the neighbourhood of the equator. At the equator 

 itself the motion of the particles vanishes. 



Eliminating u, v between (2) and (3), or putting ,9 = in Art. 

 208 (8), we find 



1 d ( hsinO 



We shall consider only the case of uniform depth (h = const.). 

 Writing //. for cos 6, the equation then becomes 



where /3 = ^ma/h= 4<ri*a 2 /gh ..................... (6). 



The complete primitive of this equation is necessarily of the form 



where </> (p), F(IJL) are even functions, and /(/&) is an odd function, 

 of //,, and the constants A, B are arbitrary. In the case of an 

 ocean completely covering the globe, it is not obvious at first sight 

 that there is any limitation to the values of A and B, although on 

 physical grounds we are assured that the solution of the problem is 

 uniquely determinate, except for certain special values of the ratio 

 /(=<7/2ft), which imply a coincidence between the 'speed' of the 

 disturbing force and that of one of the free oscillations of sym- 

 metrical type. The difficulty disappears if we consider first, for a 

 moment, the case of a zonal sea bounded by two parallels of 



