206-208] GENERAL EQUATIONS. 347 



As in Art. 200, free steady motions are possible, subject to certain con 

 ditions. Putting f=0, we find that the equations (1) and (2) are satisfied by 

 constant values of u, v, , provided 



q df a d 



a, __ _ _ &amp;lt;j_ _ _ i M y _ _ . (11) 



Zna sin 6 cos & da 2rai cos 6 d& &quot; 

 d (h sec 0, t] 



and rf( tfl .) - .............................. &amp;lt; U1 &amp;gt;- 



The latter condition is satisfied by any assumption of the form 



(iv), 



and the equations (ii) then give the values of w, v. It appears from (ii) that 

 the velocity in these steady motions is everywhere parallel to the contour-lines 

 of the disturbed surface. 



If h is constant, or a function of the latitude only, the only condition 

 imposed on is that it should be independent of &amp;lt;o ; in other words the eleva 

 tion must be symmetrical about the polar axis. 



208. We will now suppose that the depth h is a function of 6 

 only, and that the barriers to the sea, if any, coincide with parallels 

 of latitude. Assuming, further, that H, u, v, fall vary as 

 where 6 is integral, we find 



iav + 2nu cos = -&quot; (f- f) 



.., . u 1 (c (Aw sin $) 



with irf= A\~ ~TQ ~ 



Solving for u, v, we get 



d 



V = 7, ^^ 3 



If we put, for shortness, 

 these may be written 



v = 



4m(/ 2 -cos 2 6&amp;gt;)V 



/cos 0d? 

 - -T + cosec 6 



