1892.] liquid condition of the Earth's interior. 345 



that, looking backwards, the moon's orbital velocity increases very 

 rapidly. Now co is the earth's angular velocity minus the moou's 

 orbital velocity. If then retrospectively the moon's orbital velocity 

 increases more rapidly than the earth's angular velocity, co will 

 diminish. 



11 we put u = ' 



\2 aW-gh, 



, du _ ffjLCiV 2ar co (a 2 co 2 + gh) 



then dco - ~ [TJ (aW-ghf • 



Hence, so long as a 2 co 2 is greater than gh, u will increase as co 

 diminishes. Moreover the moon's distance diminishes. Hence, 

 (fj.a/2) 2 varying inversely as the sixth power of the distance will 

 increase very rapidly; so that on both these accounts the heat 

 generated in the water per lunar day will rapidly increase. It 

 must not however be forgotten that the length of the lunar day 

 increases, so that fewer of them go to a year. 



The above expression would become infinite if 



co = s/ghja ; 



that is if co = 0-000039, 



whereas at the present co = 0*00007.3. 



But such a result cannot be relied upon, because the same value 

 of co would make the expression for the height of the tide infinite, 

 whereas in the formation of the differential equation from which it 

 is found it has been assumed to be small. But it is evident that 

 the generation of heat in the water must increase as that value of 

 co is approached, and that something of the nature of a catastrophe 

 will have happened at that juncture, because, going back in time, 

 when that epoch has been passed the expression for the height of 

 the tide is found to have changed signs, and consequently high 

 and low water will have interchanged places then. 



If co is less than vgh/a, then du/dco becomes positive, and the 

 heat generated in the water rapidly diminishes as co diminishes. 



We know that "Jgh is the velocity of the free wave, with which 

 a disturbance in the water would be propagated under the influ- 

 ence of gravity alone. 



The friction of the tides against the coast-lines will of course 

 have had some effect in retarding the rotation, but how much we 

 cannot estimate*. 



"We have seen that the fact that the speed of the forced tide 

 wave in the ocean differs from that of the free wave is a cause of 



* Airy's " Tides and Waves," § 544. Encycl. Met. quoted by Sir W. Thomson, 

 *'Geol. Time." Trans. Geol. Soc, Glasgow, 1868. 



