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 



9 d ff d .... 



~ 2na sin 6 cos B cfc> ' V== 2nacos6 d6 W> 



** rfffl.) < iu >- 



The latter condition is satisfied by any assumption of the form 



and the equations (ii) then give the values of u, 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 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 e i(trt+8w+t) , 

 where s is integral, we find 



* M ' . (3)) 



with ia = ~ 



asm 



Solving for u, v, we get 

 iala 



cot )(-), 



' ...(5). 



If we put, for shortness, 

 these may be written 



^ |...(7). 

 cosec " 



