Deep-sea Tides, and the Effect of Tidal Friction. 175 

 And 



tan 28= ^= fc > 



whence 



d . tan 26, or -. — . g g = — -7=d • 



.v 



(cos2S) 2 "2v/^ '*-&■ 



2v/^ L2v/K(/c-K) («-K)»J 4? (k-K) 2 * 



Now, when the ridges coincide, the distance e will be equal 

 to that by which the luminaries have separated in the time t, or 



5 t. Whence, substituting*, we obtain 



: '^^t^fe '■;' ;■;,-. ~ 



This is independent of both h and i; so that for small and slow 

 changes of force, the lagging of the maximum tide behind the 

 time of maximum force depends only on the mean rate m, a 

 more rapid change of force being counterbalanced by a greater 

 value of e. 



For numerical calculation we may make our equations for 8 

 and for t homogeneous by reintroducing the radius (a) of the 

 earth, observing that t and /are numbers ; whence we see that 



fa* 

 generally t is of the order of magnitude of J -=^ i and may be very 



sensible. 



' And this completes our theory of the tide in an equatorial 



canal. 



8. If we suppose a canal drawn in any way along the earth's 

 surface, the problem becomes very complicated in its details, 

 and I do not here propose to work it out; but the principles 

 of the past investigation are generally applicable — unless, in- 

 deed, we have to deal with a case where the length of wave 

 necessary to keep pace with the changes in any part of the 

 moon's force ceases to be much larger than the depth of the sea 

 (as in a small circle close to the pole, for instance), and so have 

 to take account of vertical effective forces. 



For, first, with the motions restrained to be longitudinal only, 

 the variations in the centrifugal force (mentioned in 4 (a)), 

 although they are not vertical as on the equator, have no effect; 



Referring any point in the canal to its polar distance (6) and 

 longitude (©), if ds be an element of the canal, the velocity rela- 

 tively to it of a particle of water and the components in and, 



