126 THE TIDAL PROBLEM. 



We may use equation (140) to compute the time it — to) corresponding to 

 any value of P provided the physical condition of the system, and partic- 

 ularly that of the earth, has not changed sensibly in the mean time. The 

 data furnished by geology are showing more and more that the earth has 

 been sensibly in its present state, except for approximately periodic oscil- 

 lations in its climate, for many millions of years. For the purposes of 

 computation we shall assume that it has been indefinitely so. While this 

 assumption is not strictly true, the actual observational data show that it 

 IS almost certainly much less in error than the assumption, stimulated by 

 the Laplacian theory of the origin of the earth, that our planet was fluid 

 in the not very remote past. Remembering the fact that we are assuming 

 simply that the apparent secular acceleration of the moon's mean motion 

 is due to tidal friction and that it is a measure of the rate of tidal evolution, 

 and that we are assuming further that the physical condition of the earth 

 and moon has not changed in the time covered by our calculations, we find 

 from (23) that Avhen D = 20 hours the value of P was 24.096 days, and from 

 (140) that P had this value 220,700,000,000 years ago. If the action of 

 the sun had been included the interval Avould have been decreased by about 

 20 per cent. It is impossible to believe that the neglected factors, such as 

 the eccentricity and inclination of the moon's orbit, could reduce the time 

 enough to change the order of these results. This computation, which has 

 the merit of being based quantitatively on actual observations, points very 

 strongly to the conclusion that tidal evolution is so slow a process that it 

 can not have played an important role in the earth-moon system, even when 

 we consider an interval of a billion years. 



There is, however, another possibility that may be considered. It is at 

 least conceivable that there may be unknown forces acting upon the earth- 

 moon system in such a way that they largely mask the relative secular 

 tidal acceleration of the moon. Any thing increasing the moon's distance 

 and period without otherwise disturbing its motion, or any thing acceler- 

 ating the rotation of the earth, would tend to offset the secular acceleration 

 produced by tidal friction. The possible secular contraction of the earth 

 is a factor working in the right direction. But from the numbers obtained 

 above it follows that, if we are to escape from the conclusion that tidal fric- 

 tion is now a negligible factor, we must assume that the actual relative tidal 

 acceleration of the moon is several hundreds of times 4" per century. Sup- 

 posing the reduction to 4" is due to the acceleration of the rotation of the 

 earth because of shrinking, it follows that at every epoch in the past the 

 day and the month were more nearly equal than they would have been 

 except for this factor. Finally, at the limit at which they were equal, 

 their common period was many times that computed above, and their 

 great initial distance fatal to the fission theory. If there are unknown 

 forces retarding the moon's revolution, the conclusions are the same. 



