610 Proceedings of the Royal Irish Academy. 



Peopositioi^ II. 



It is required to find the rate oflengtheniny of the sidereal day, caused 

 hy the displacement of the tidal spheroid hy friction ; under the same 

 conditions as those stated in the last Proposition. 



"When there is no friction, the minor axis of the tidal spheroid 

 points to the moon, and the major axis is at right angles to the moon's 

 direction. 



In the case of an equatorial canal, uith the moon al"ways in the 

 equator, if /denote the coefficient of friction, supposed proportional to 

 the relative velocity of the moving water, the velocity of the water is 

 represented by 



]c [ f . \ 



u = Vo + — cos 2m sin 2m . (/) 



2a) Y 2oj / 



The periodic term vanishes, not when m = 45°, but when m = 45° + a;; 

 where x is found from the equation 



/ 

 tan 2a; = - ^--. 

 2o) 



The tidal spheroid is. therefore, displaced through an angle x, in a 

 direction opposite to the earth's rotation, and the phases of the tide at 

 all places are accelerated. If ^ be the complement of x, the major axis 

 of the tidal spheroid will form an angle ^ with the direction of the 

 moon, and the two caps of water lying between the ellipse and circle 

 already described will give rise, by the action of the moon and anti- 

 moon, to a retarding couple, which tends to lengthen the day. 

 The magnitude of this couple is 



hn X sin 2<^ x a, 



where m is the mass of water lying between the ellipse and circle ; or, 



Retarding couple - 8G = irwha^s.^ sin 2^. 



This couple is of the first order with regard to eo) whereas the re- 

 tarding couple produced by the residual tidal current was of the second 

 order only. 



Hence we have, as before, 



Sr . + 3 X 3-06 xyW. Co sin 29^ ^^^ 

 Stt X 5"5 a- 



