536 On the Tidal Retardation of the Earth's Rotation. 



the earth is exposed render it a very untrustworthy time-keeper. 

 For instance, let us suppose ice to melt from the polar regions 

 (20° round each pole, we may say) to the extent of something 

 more than a foot thick, enough to give 1*1 foot of water over 

 those areas, or *066 of a foot of water if spread over the whole 

 globe, which would in reality raise the sea-level by only some such 

 almost undiscoverable difference as | of an inch, or an inch. This, 

 or the reverse, which we may believe might happen any year, 

 and could certainly not be detected without far more accurate 

 " observations and calculations for the mean sea-level than any 

 hitherto made, would slacken or quicken the earth's rate as a time- 

 keeper by one-tenth of a second per year*. 



Again, an excellent suggestion, supported by calculations which 

 show it to be not improbable, has been made to the French Aca- 

 demy by M. Dufour, that the retardation of the earth's rotation 

 indicated by M. Delaunay, or some considerable part of it, may 

 be due to an increase of its moment of inertia by the incorpora- 

 tion of meteors falling on its surface. If we suppose the pre- 

 vious average moment of momentum of the meteors round the 

 earth's axis to be zero, their influence will be calculated just as 

 I have calculated that of the supposed melting of ice. Thus 

 meteors falling on the earth in fine powder (as is in all probabi- 

 lity the lot of the greater number that enter the earth's atmo- 

 sphere and do not escape into external space again) enough to 

 form a layer about ^ of a foot thick in 100 years, if of twice 

 the density of water, would produce the supposed retardation of 



any approach to an exact determination of the amount of the actual re- 

 tardation of the earth's rotation by tidal friction, except by extensive and 

 accurate observation of the amounts and times of the tides on the shores of 

 continents and islands in all seas, and much assistance from true dyna- 

 mical theory to estimate these elements all over the sea. But supposing 

 them known for every part of the sea, the retardation of the earth's rota- 

 tion is easily calculated by quadratures. 



* The calculation is simply this. Let E be the earth's whole mass, a 

 its radius, k its radius of gyration before, and V after the supposed melting 

 of the ice, and W the mass of ice melted. Then, since fa 2 is the square of 

 the radius of gyration of the thin shell of water supposed spread uniformly 

 over the Avholej surface, and that of either ice-cap is very approximately 

 |a 2 (sin20°) 2 , we have 



Efc' 2 =E& 2 +Wa 2 [f-| (sin20) 2 ]. 

 And, by the principle of the conservation of moments of momentum, the 

 rotatory velocity of the earth will vary inversely as the square of its radius 

 of gyration. To put this into numbers, we take, as above, k' 2 = 3 l a? and 

 a= 21 X 10 6 . And as the mean density of the earth is about 5} times that 

 of water, and the bulk of a globe is the area of its surface into i of its 

 radius, 



E : W : : 5l 5 - : -066. 

 o 



