114 THE EVOLUTION OF SATELLITES. 



edge of the laws of the moou's motion is not yet quite accurate enough 

 for the absohitely perfect calculation of eclipses which occurred many 

 centuries ago. In this way it is known that within historical times the 

 retardation of the earth's rotation and the recession of the moon have 

 been, at any rate, very slight. 



It does not follow from this that the changes have always been 

 equally slow, and indeed it may be shown by mathematical arguments 

 that the efficiency of tidal friction increases with enormous rapidity as 

 we bring the tide raising satellite nearer to the planet. The law of tidal 

 friction is that it varies according to the inverse sixth power of the 

 distance; so that with the moon at half her present distance, the rate 

 of retardation of the earth's rotation would be sixty- four times as great 

 as it now is. Thus, although the action may now be almost insensibly 

 slow, yet it must have proceeded with much greater rapidity when the 

 moon was nearer to us. 



There are many problems in which it would be very difficult to follow 

 the changes in the system according to the times of their occurrence, 

 but where it is possible to banish time, and to trace the changes them- 

 selves in due order, without reference to time. In the sphere of com- 

 mon life, we know the succession of stations which a train must pass 

 between New York and Boston, although we may have no time-table. 

 This is the case with our astronomical problem ; for although we have 

 no time-table, yet the sequence of the changes in the system may be 

 traced accurately. 



Let us then banish time, and look forward to tlie ultimate outcome of 

 the tidal interaction of the moon and the earth. The day and the 

 month are now lengthening at relative rates which are calculable, 

 although the absolute rates in time are unknown. It will suifice for a 

 general comprehension of the problem to know that the present rate of 

 increase of the day is much more rapid than that of the month, and 

 that this will hold good in the future. Thus, the number of rotations 

 of the earth in the interval comprised in one revolution of the moon 

 diminishes; or, in other words, the number of days in the month dimin- 

 ishes, although the length of each day increases so rapidly that the 

 month itself is longer than at present. For example, when the day 

 shall be equal in length to two of our actual days, the month may be 

 as long as thirty-seven of our days, and then the eartli will spin round 

 only about eighteen times in the month. 



This gradual change in the day and the month proceeds continuously 

 antil the duration of a rotation of the earth is prolonged to fifty-five of 

 our present days. At the same time, the month, or the time of a revo- 

 lution of the moon around the earth, will also occupy fifty-five of our 

 days. Since the month here means the period of the return of the moon 

 to the same place among the stars, and since the day is to be estimated 

 in the same way, the moon must then always face the same part of the 

 earth's surface, and the two bodies must move as though they were 



