Deep-sea Tides, and the Effect of Tidal Friction. 179 



only one, and then giving to y, u, and v the sums of the three 

 corresponding values. 



Making a> r =0 } or referring our longitudes to the meridian on 

 which the moon is at any time, we will write for HP 2 , 



A{f cos 2 0— 1}+Bsin20cosft> + Csin 2 0cos2a). 



The first term is independent of co, and gives no motion, but 

 a permanent change (while we treat 8 as constant) in the figure 

 of the sea : in this case the first equation is alone significant, and 

 gives 



or 



<,y = A(fcos 2 0-i). 



"When the moon is on the equator the force is represented by the 

 last term alone ; whereas neither of the other terms can subsist 

 alone in nature. We will therefore now take this case, the full 

 discussion of which will lead us on to the other. 



We want a solution independent of g>, so that the wave may 

 always have the same aspect as seen from the meridian of the 

 moon. If, now, we try to make the wave in each parallel of lati- 

 tude follow, as nearly as may be, the laws of a wave in an equa- 

 torial canal, and make y=y x cos2o> and v =v 1 cos2g>, «/, and v x 

 being functions of only (and therefore constants for a given 

 latitude), we see we shall succeed if we make u=u 1 sm2co. & 

 will then disappear from all the equations, and we shall have, 

 after reductions, 



C sin cos 0— \g ~ = n sin cos 0v l ~mu i) 



C sin + 4^ =m sin 0v, — rccos 0u u 

 sin * " 



1 d./c sin n 

 sm0 W 



—2my l 



- 1 -2/^ 



d/c , fdu x , COS0 „ I 



(I) 



If we knew the law of depth, we should use these equations 

 to find the wave ; and Laplace and Airy have accordingly as- 

 sumed a law of depth, and so proceeded. I think if we may 

 hope that these methods will be so extended as to be applicable 

 to actual facts, this will be by obtaining from observation the law of 

 tide in different latitudes and thence deducing the law of depth, 



N 2 



