Beep-sea Tides ; and the Effect of Tidal Friction, 173 



m <2 =g'K ) we get 



{(^) 



V £K"U =H . or c= .^fr? 



4^ J |%(«-K)^+/ S K; 



and by division, 



tan 28=— &—- *«/* 



2g{K-K) 2y/g(tc-K) 



IT 



With shallow seas tan 28 is negative, and S> -r, or the pas- 

 sage of the moon over the meridian is nearer to the time of low 

 water which has preceded it than to that of high water which 

 follows it. With seas deeper than K, the reverse would be the 



rrr 



case. When tc — K, 8=—, and the passage is at midtide or 

 slack water. In this case c= ^ ? H, or -j- H. m is here 



J' y 



supposed to be expressed in angular measure, and is about 



1 c 



., - „• tf therefore cannot be larse. unless f is extremely 

 15,000 H ° ' J J ~ 



minute. 



7. If there be two bodies moving over the equator at differ- 

 ent rates (m v m^j and exerting different forces, the conditions 

 will be fulfilled by superposing the wave suited to one body upon 

 that which suits the other; i. e., if y l be the elevation at a> due 



to one body (and therefore, by what precedes, — y x the velocity 



dy K 



and A-~ the form of the corresponding force), then, with the 



two bodies, y=yi + y^ aud v= — — ^-^ will satisfy the equa- 



fC 



tions, whether taking account of friction or not. 



This gives us the rule for combining the lunar and solar tides. 



Bat it also enables us to deal with the fact that neither lumi- 

 nary maintains a constant equatorial force on the canal. We 

 are only concerned with the force estimated in the direction of 

 that canal ; and whether the luminary slowly alters its declina- 

 tion or its distance from the earth, the result is a periodical 

 change in the intensity, but not in the form of the force; and 

 the change may be represented by substituting for the constant 

 H an expression H + A cos i t } or, if necessary, by adding a series 

 of terms similar to the second with different values of i. H then 

 corresponds to the mean disturbing force, and H+A to the 

 maximum and minimum, and the time t is reckoned from the 

 time of maximum force. 



