Deep-sea Tides, and the Effect of Tidal Friction. 185 



the fact that the mass of the solid earth must move with the 

 protuberances without sliding*. 



The Astronomer Royal having (by an oversight, I venture to 

 think) convinced himself that no effect on the earth's rotation can 

 be produced by those forces of the first order, with friction, of 

 which his own canal theory took account, seeks for such an effect 

 entirely in a current produced by forces of the second or higher 

 orders. 



In an extract from the 'Rede Lecture' (May 1866) given in 

 this Journal, Sir W. Thomson calculates the effect in the mode 

 which Mayer also proposes, as if the water were merely a 

 machine for conveying the whole force of the moon to the solid 

 earth ; but this extract contains no discussion of the principle. 

 I proceed to give my own present view of the matter. 



The only case in which the tide with friction has been inves- 

 tigated is in a canal. For the purpose of facilitating the calcu- 

 lation, we have assumed that the friction at the bottom of each 

 vertical column acts as an equable accelerating force on every 

 particle of water in that column, i. e. that it is equivalent to a 

 set of moving forces whose resultant is proportional to the mass 



of the column, or (/c + y)da), acting at a height ^ -f - above 



the bottom. If this be physically true, the reaction on the 

 solid earth is equal and opposite to this. The accelerating force 



isfvj or J — - (C), and the moment to turn the earth westward 



is —^(ic + y) (l+ -9- + ^)^; ana * as changing the sign of y 



changes the numerical value of this expression, the sum of the 



moments all round the equator is not zero as Mr. Airy has 



made it. Putting y — c cos 2co, and integrating from co to 27r + ay, 



the only term which does not vanish is that arising from (cos 2o)) 2 , 



_ . . fmc 2 ... N 

 and it gives ir ■ [L+te). 



Now, treating the question as if the protuberances were adhe- 

 rent to the solid earth, the moon's horizontal moving force on the 

 elevation (y) above the mean level is, by our formulas, 

 — 2#H sin 2 (a> — h) ydco ; 



and its moment round the centre of the earth is 1-f k-\- ¥-, 



Putting its value for y and integrating, the only term which 



does not vanish is that arising from sin 2(<y — 8) cos 2o>, and it is 



fmc 

 2gUc sin 28(1 + *)7r, which (by 6) =^ (1 + k)7t, the same 



* I made some very crude and blundering remarks on extracts from 

 this paper, which alone I had seen. 



