338 Rev. 0, Fisher, On the hypothesis of a [May 30, 



In forming the potential of the protuberance of the "bodily tide 

 the earth has been taken as homogeneous. But the superficial 

 parts having only half the mean density, it seems that the value 



& aft 1 



of - — ought to be taken at one half that assumed, and then 



^^?= 0-19894, 

 5 T 



and we find for the tide 



- H x 0-80106 cos (2 (<£ -cot)-D 



which shows that it will be diminished by only ^th of what its 

 height would be if the earth was rigid. 



We learn from this expression that high ocean tide will occur 



7T D 



when 2 (<£ — cot) — D = ir, that is when <£ — cot = -= + -~ to the 

 west of the moon ; and high earth tide will occur when <£ — cot = — -= , 



or - to the east of the moon. Hence the crests of the ocean and 



2 



earth tides are separated by the obtuse angle -= H = — , so that 



the tidal protuberances of both of them, which are nearest to the 

 moon, are to the east of it ; and the effects of the couples caused 

 by the moon's attraction upon both of them will be to retard the 

 earth's rotation. 



Prof. Darwin remarks, that the expression for the height of the 

 ocean tide as affected by the bodily tide is subject to a modifica- 

 tion of the same form on the equilibrium theory as on the canal 

 theory, with the exception of a change of sign. Hence on that 

 theory also, which neglects the inertia of the water, and therefore 

 less nearly represents the case of nature, the ocean tide would be 

 diminished by the same factor, and therefore only to the small 

 extent of about one-fifth, as has been now shown would be the 

 case on the canal theory. 



The lag of the bodily tide has here been put at 14', because 

 Darwin has shown* that, on the hypothesis of approximate 

 liquidity, the reaction of the moon on the bodily protuberance 

 with that amount of lag would account for the unexplained 

 acceleration of the moon's mean motion at the present time of 

 4 seconds in a century. It is evident that if the lag of the bodily 

 tide is larger, cos 2e will be smaller, and the reduction of the tide 

 in the canal will be still less. 



' * "Precession of a viscous spheroid,'' § 1-4. 



