356 TIDAL WAVES. [CHAP. VIII 



investigation*. We shall find that it enables us to calculate the 

 forced oscillations when the depth follows the law 



h = (I-qcos*0)h Q ........................ (2), 



where q is any given constant. 



Taking an exponential factor e i(nt+ta+e) , and therefore putting 

 s = l, /= ^, in Art. 208 (7), and assuming 



f = Csin0cos&amp;lt;9 ........................ (3), 



we find u= i&amp;lt;rC/m, v = a-C/m.cos0 ............... (4). 



Substituting in the equation of continuity (Art. 208 (4)), we get 



ma 

 which is consistent with the law of depth (2), provided 



C=- - H &quot; 



mi 2qhJma $ /h _ x 



This gives f=- Y ; f ..................... (7). 



1 



One remarkable consequence of this formula is that in 

 the case of uniform depth (q = 0) there is no diurnal tide, so 

 far as the rise and fall of the surface is concerned. This result 

 was first established (in a different manner) by Laplace, who 

 attached great importance to it as shewing that his kinetic theory 

 is able to account for the relatively small values of the diurnal 

 tide which are given by observation, in striking contrast to what 

 would be demanded by the equilibrium-theory. 



But, although with a uniform depth there is no rise and fall, 

 there are tidal currents. It appears from (4) that every particle 

 describes an ellipse whose major axis is in the direction of the 

 meridian, and of the same length in all latitudes. The ratio of the 

 minor to the major axis is cos 6, and so varies from 1 at the poles 

 to at the equator, where the motion is wholly N. and S. 



213. Finally, we have to consider Laplace s Oscillations of the 

 Third Species, which are such that 



f = J ff&quot; / sin a 0.cos(&amp;lt;r$ + 2G&amp;gt; + ) ............... (1), 



* Taken with very slight alteration from Airy (&quot;Tides and Waves,&quot; Arts. 95...), 

 and Darwin (Encyc, Britann., t. xxiii., p. 359). 



