Deep-sea Tides, and the Effect of Tidal Friction, 177 



presented by a superficial wave-form whose intersection with the 

 line of the canal^gives a pressure-force ; to each such imaginary 

 wave must correspond a tide-wave of the same form, with an ab- 

 solute height such as will adjust the rate of propagation to that of 

 the moon ; and the sum of these waves will represent the true tide. 



I should add that I have not worked out any of the cases in 

 Airy in this way, and do not know whether they would come 

 out more or less simply than they do there. There is no differ- 

 ence in principle in the methods ; and this seemed to me the 

 best introduction to what follows. 



9. We now pass to Laplace's problem — to find the conditions 

 under which we may have in an open sea covering the globe a 

 permanent wave-shape accompanying the moon (which we will 

 suppose to move uniformly in longitude) , with the same angular 

 velocity (m). 



If y be the height, u and v the southerly and westerly (angular) 

 velocities, we must have, by our former reasoning, 



dy __ dy du _ du dv _ dv .pv 



dt "" da) dt dco dt dco' 



"We will also assume, with Laplace, that the vertical motions 

 may be neglected. This is so, by our former reasoning, where- 

 ever the linear length of the wave is large compared with k. 

 But it does not seem clear that near the pole the vertical motions 

 may not be comparable with the horizontal ; and this may per- 

 chance account for a difficulty we shall hereafter come upon. 



On this assumption, however, it follows that the pressure-force 



du 

 of the wave will be that due to the weight only, and be — g -p 



du 

 in the direction of any line s (and therefore —g ~ southward, 



and — g-. — -^-r- westward), and that we may suppose all the 



particles in the same vertical column to move with the same 

 velocity. 



But there will now be an uncounteracted pressure arising 

 from the centrifugal force (8), which estimated southward is 

 ~2rasin cosOv. 



And in forming our equation for the westerly accelerating 

 force we must not equate it to the rate of apparent change of 



relative velocity — jr~> uut to tn ^ s quantity diminished by 



= 2ncos6-j-), because (8) thus much of the apparent 



increase is due to its not being pressed eastward, but maintain- 

 ing its own angular rotation while moving southwards. 

 Phil. Mag. S. 4. Vol. 33. No. 222. March 1867. N 



