340 TIDAL WAVES. [CHAP. VIII 



where # refers to the proper asymptote of the curve (iv). This gives the 

 speed of any free mode in terms of that of the corresponding mode when 

 there is no rotation. 



204. As a sufficient example of forced oscillations we may 

 assume 



(17), 



where the value of a- is now prescribed. 



This makes V 2 f = 0, and the equation (4) then gives 



f=4/-,(*,r)e &amp;lt;*+&quot;+&amp;gt; .................. (18), 



where A is to be determined by the boundary-condition (7), 

 whence 



.c ......... (19). 



d ,, , . 2sn 



This becomes very great when the frequency of the disturbance 

 is nearly coincident with that of a free mode of corresponding 

 type. 



From the point of view of tidal theory the most interesting cases are 

 those of s = l with o-=n, and s = 2 with o- = 2rc, respectively. These would 

 represent the diurnal and semidiurnal tides due to a distant disturbing body 

 whose proper motion may be neglected in comparison with the rotation n. 



In the case of s = l we have a uniform horizontal disturbing force. 

 Putting, in addition, &amp;lt;r = n, we find without difficulty that the amplitude of 

 the tide-elevation at the edge (r = a) of the basin has to its equilibrium-value 

 the ratio 



where = JV(3j8). With the help of Lodge s tables we find that this ratio has 

 the values 



1-000, -638, -396, 



for/3= 0, 12, 48, respectively. 



When o- = 2?i, we have K = 0, /,(*, r) = r 8 , and thence, by (17), (18), (19), 



i.e. the tidal elevation has exactly the equilibrium -value. 



This remarkable result can be obtained in a more general manner; it 

 holds whenever the disturbing force is of the type 



provided the depth h be a function of r only. If we revert to the equations 



