192 



Mr. G. H. Darwin on 



[Dec. 19, 



apparent acceleration of 4" per century, and I found that, if the earth 

 were purely viscous, the moon must be undergoing a secular retarda- 

 tion of 3'6" per century, while the earth (considered as a clock) must 

 be losing fourteen seconds in the same time. The obliquity also must 

 be diminishing at the rate of 1° in 470 million years. 



Under these circumstances the earth must have so great an effective 

 rigidity that the bodily semi-diurnal and diurnal tides would be quite 

 insensible ; the bodily fortnightly tide would however be so consider- 

 able that the oceanic fortnightly tide would be reduced to one-seventh 

 of its theoretical value on a rigid nucleus, and the time of high water 

 would be accelerated by three days. 



The supposition that the earth is a nearly perfectly elastic spheroid 

 leads to very different results in this respect, which, however, I will 

 now pass over. 



From this and other considerations, I conclude that a secular accele- 

 ration of the moon's motion affords no datum for determining the 

 present amount of tidal friction. 



Sir W. Thomson has discussed the probable age of the earth from 

 considering the tidal friction, and he derived his estimate of the rate 

 at which the earth's diurnal rotation is slackening, principally from 

 the secular acceleration of the moon. He fully admitted that his data 

 did not admit of precise results, but, if I am correct, it certainly 

 appears that his argument loses some of its force. 



The differential equations, which have to be solved in order to 

 investigate the secular changes in the configuration of the three bodies, 

 are exceedingly complex, and I was only able to solve them by a labo- 

 rious method, depending partly on analysis and partly on numerical 

 quadratures. 



The solution was only applicable to the case where the earth is a 

 purely viscous body, and the numerical value chosen for the coefficient 

 of viscosity was such that the changes proceed with about the maxi- 

 mum rapidity. Starting with the present values of the obliquity, day, 

 month, and year, the changes were traced backwards in time. As we 

 go backwards we find the year sensibly constant, but the obliquity, 

 day and month all diminishing — the last with far the greatest rapidity. 

 The changes proceed at a rapidly increasing rate, as in the retrospect 

 the moon approaches the earth. 



At the point where I found it convenient to stop in the first method 

 of solution, about 56 million years have been traversed backwards, 

 and the obliquity is found to have diminished by 9°, the day is found to 

 have fallen to 6 hrs. 50 mins., and the sidereal month to only 1 day 

 14 hrs. 



It is a question of great interest to geologists to determine whether 

 any part of changes of this kind can have taken place during geological 

 history ; and I conclude that it might be so. The physical meaning 



