£)eep-sea Tides, and the Effect of Tidal Friction. 171 



minishing proportionally the effective horizontal force. There- 

 fore by proper adjustment of the height of the wave (c) we can 



nil 



make the proportion of the joint force to ~ exactly that which 



is necessary to start and keep the wave moving at the same rate 

 with, and so always remaining under, the moon. 



If m be the moon's apparent rate, and if K be that value of k 

 which makes gK = m 2 , K is about twelve miles. If, then, the canal 

 be of less depth than this, c must be negative, or it is low water 

 under the moon ; if deeper, then high water. 



Let H be the height of a wave of the necessary form which 

 would exactly represent the moon's force, i. e. let its height at ft> 



3M 

 be — H cos 2o), where H= 7^3-. Then, the equation to the 



actual tide-wave being y = ccos2a>, the joint horizontal force is 



-*{■-?}- 



dco 

 and we must make 



c cos 2<y 



whence 



^l^ 1 " 7 jfc = m?=gli, 



K 



H, 



which is positive or negative according as k is greater or less than 

 K, in accordance with what has been said above. 



When k=K, or the free wave would have the moon's velocity, 

 e becomes infinite ; and our formulas, which are approximations 

 on the supposition of c small compared with k, fail before this 

 point is reached. The explanation is simple. Eor, as H is given, 



Ti- 

 the ratio — can only diminish by the increase of c; the less the 



influence of the moon required to alter the rate of propagation,' 

 the greater must be the wave. 



6. In all that has been said above, the consideration of fric- 

 tion has been omitted. ! 



The losses of mechanical power by friction or by impact seem 

 to be identical or strictly analogous in their causes ; and I con- 

 ceive that such loss must occur in the tide-wave, not only by 

 reason of the friction at the sea-bottom retarding the lowest 

 stratum, and that stratum the next above, and so on, but also by 

 the series of infinitesimal collisions between contiguous columns 

 moving with different velocities. This last effect, however, will 



