132 THE TIDAL PROBLEM. 



as predicted by the gravitational theory neglecting tidal evolution. This 

 relative gain in longitude, if established by observations, will give us the 

 measure of the rate of tidal evolution at the present time, and we can safely 

 apply it, by properly varying the factors depending upon the moon's dis- 

 tance, over any interval during which the physical condition of the earth 

 has been essentially as it is at present. Since the geological data go to show 

 that the physical state of the earth has been about as it is now for many 

 tens of millions of years, and do not give certain evidences of any radically 

 different general physical conditions, we are perhaps justified in boldly 

 applying results based on the present rate of gain of the moon for a very 

 long interval of time. 



It is well known that a comparison of ancient and modern eclipses shows 

 that the moon has an acceleration in longitude of about 4" per century 

 which is not explained by perturbations. Let us assume that this is due 

 to tidal friction and is the measure of it at the present time. At this rate 

 it will take over 30,000,000 years for the moon to gain one revolution. 

 Consequently we see without any computation that it must have been an 

 extremely long time in the past when its period was a small fraction of its 

 present period. 



The problem was treated in section XV, and it was found there that, if 

 the physical condition of the earth has been essentially constant, the length 

 of the day was 20 of our present hours, and of the month, 24 of our present 

 days not less than 220,000,000,000 years ago. It is extremely improbable 

 that the neglected factors, such as the eccentricity of the moon's orbit, 

 could change these figures enough to be of any consequence. This remark- 

 able result has the great merit of resting upon but few assumptions and 

 in depending for its quantitative character upon the actual observations. 

 If it is accepted as being correct as to its general order, it shows that tidal 

 evolution has not affected the rotation of the earth much in the period 

 during which the earth has heretofore been supposed to have existed even 

 by those who have been most extravagant in their demands for time. 

 And if one does not accept these results as to their general quantitative 

 order, he faces the embarrassing problem of bringing his ideas into harmony 

 with the observations. 



If tidal friction has been an important factor in the evolution of the 

 earth-moon system, then presumably it has also been an important factor 

 somewhere else in the universe. Certainly one would expect to find the 

 theory encountering no difficulties in the case of any other planet. But 

 the place where it has been appHed most is in the orbits of the binary stars, 

 which have been supposed to have become binaries through fission and 

 to have become widely separated as a consequence of tidal evolution. The 

 first difficulty is, as has been pointed out, that there must have been an 

 initial eccentricity of the relative orbit, and this eccentricity would cause 

 the bodies to reunite as a consequence of tidal friction. But there is another 

 important difficulty, as was explained in section IX. If we suppose the 

 binary to be composed of two equal suns moving just after separation 

 as a rigid body, and if we waive the effects of the eccentricity for the sake 

 of the argument, we find that there is a maximum distance to which the 



