110 On the Theory of the Fortnightly Tide. 



polar coordinates Laplace's equations are 



dv . /i '/ ^ - 



- 7 -- + 2m<cosfl= — - . (g— 0. . • • (10) 



dv . rt „ <7 <rZ 



d 



*T_ 1 f d(7iuzm0) d(hv)-} 



dt asm 0.\. dd day )' ' ' ^ } 



-where tu v are the velocities along and perpendicular to the 

 meridian, f is the elevation at any point, f the equilibrium 

 value o£ £, a denotes the earth's radius, a the angular velocity 

 of rotation, and h the depth of the ocean at any point. To 

 determine the free steady motions, we are to put £=0 as 

 well as du/dt, dv/dt, dfydt. Thus 



9 dl g d$ 



2/ia sin 6 cos 6 dco' 'ina cos 6 dO ' 



and 



d(7isec0)dS d(h sec 0) d^ 



(12) 



dd dco dm d6 



= 0. ... (13) 



If li sec 6 be constant. (13) is satisfied identically. In any 

 other case a restriction is imposed upon f. If h be constant 

 or a function of the latitude only, f must be independent 

 of co ; in other words the elevation must be symmetrical 

 about the polar axis. In correspondence therewith u must 

 be zero and v constant along each parallel of latitude. 



In the application to an ocean completely covering the 

 earth, such as is considered in Darwin's solution, the above 

 conditions are easily satisfied, and the free steady motions, 

 thus shown to be possible, explain the large deviation of the 

 calculated fortnightly tide from the equilibrium value. What 

 does not appear to have been sufficiently recognized is the 

 extent to which this state of things must be disturbed by 

 the limitations of the actual ocean. Since v must be con- 

 stant along every parallel of latitude, it follows that a single 

 barrier extending from pole to pole would suffice to render 

 impossible all steady motion : and when this condition is 

 secured a tide of sufficiently long period cannot deviate from 

 its equilibrium value. Now the actual state of things corre- 

 sponds more nearly to the latter than to the former ideal. 

 From the north pole to Cape Horn the barriers exist ; and 

 thus it is only in the region south of Cape Horn that 

 the circulating steady motion can establish itself. It would 

 seem that this restricted and not wholly unobstructed area 



