606 Proceedings of the Royal Irish Academy. 



be placed in an equatorial canal, extending ten degrees on each side, 

 north, and south of the equator, such an equatorial canal will be 

 about ten miles in depth. 



The velocity of the water produced in such a canal, by the earth's 

 rotation and action of the moon, is 



M = Vo + z^ cos 2m, (A) 



2(1) 



where Vf, = equatorial velocity of the earth's surface, 



3 M r 3M a r _ 1 r 

 ^2^'E^2E^'Ba~ 400,000 a ' 



277 



m = moon's hour angle ; 



a = mean value of r. 



If the alteration in the shape of the water caused by the moon's 

 action be neglected, and r considered always equal to a, then equa- 

 tion (-4) summed all round the equator will give 



for the periodic term will disappear, and no effect will be produced by 

 the moon in retarding or accelerating the rotation of the earth. If, 

 however, we take account of the altered shape of the water, we have 



T 



-= (1 - e cos^m), (-5) 



(I 



where 



c - a 



e = 



c and a being the greater and lesser semiaxes of the tidal spheroid. 

 The periodic term in {A) thus becomes 



tidm = — |f 1 - - I cos 2mdm - - (1 + cos im) dm> , 



which, when summed all round the circumference, leaves a 



Residual Current = - — x — . (6') 



