On the Theory of the Fortnightly Tide. 137 



value of the fortnightly tide being 



?=H'(i- M »), 



the actual tide for a depth of 7260 feet is found to be 



f/H' =-1515 -1-0000 p 2 +1*5153 /* 4 - 1-2120 **« 



-•2076 jn 10 + -0516 yu, 12 --0097 /x 14 + -0018 /u 16 -'0002 /* 18 , 

 whence at the poles (fi= +1) 



r=-|H'x-154, 

 and at the equator (/-t = 0) 



Again, for a depth or 29040 feet, we get 



f/H' = -2359-l-000^ 2 + -5898^ 4 

 - -1623 yu, 6 + -0258 fi 8 - -0026 ^ 10 + *0002 //, 12 , 



making at the poles 



?=-§H'x-470, 



and at the equator 



f=iH , x-708. 



It appears that with such oceans as we have to deal with 

 the tide thus calculated is less than half its equilibrium 

 amount. 



The large discrepancy here exhibited leads Darwin to 

 doubt whether "it will ever be possible to evaluate the 

 effective rigidity of the earth's mass by means of tidal 

 observations." 



From the point of view of general mechanical theory, the 

 question at once arises as to what is the meaning of this con- 

 siderable deviation of a long-period oscillation from iis 

 equilibrium value ? A satisfactory answer has been provided 

 by Lamb * ; and 1 propose to consider the question further 

 from this point of view in order to estimate if possible how 

 far an equilibrium theory may apply to the fortnightly tides 

 of the actual ocean. 



The tidal oscillations are included in the general equations 

 of small vibrations, provided that we retain in the Latter the 

 so-called gyrostatic terms. By a suitable choice of coordi- 

 nates, as in the usual theory of normal coordinates, these 



* Hydrodynamics, §§ 196, 198, 207. 



