116 THE TIDAL PROBLEM. 



XI. APPLICATION OF CASE X TO THE EARTH- 

 MOON SYSTEM. 



Observations show that in the case of the moon's orbit e = 0.0549. 

 Then we find from (98) that in this case 



M = 3.63082 £; = 1.61057 



Consequently for the moon at present the curves of fig. 15 are yet open at 

 infinity, and so far as this discussion goes, the eccentricity may increase 

 to unity, and the system be wrecked by a coUision of the earth and moon. 

 The most interesting question relates to the least possible distance of 

 the earth and moon from each other. Suppose at the time of the assumed 

 separation the eccentricity of the orbit was very small, as apparently it 

 must have been if the bodies separated by fission. Then the configuration 

 at the time of separation corresponds to the point A of fig. 15. The abscissa 

 of this point is the smaller real root of equation (104), which for the present 

 value of M we find to be 



Do = 0.206008 days = 4.944 hours 



From the relation between the period of revolution and the distance we 

 find that the distance corresponding to this period is Rq = 9,214l.O miles. 

 Neglecting the eccentricity of the moon's orbit we found for the initial 

 distance 9,194.4 miles. 



But, as was explained in the preceding section, the initial eccentricity 

 could not have been zero for the present eccentricity is different from zero. 

 The larger it was the shorter the initial period and the smaller the initial 

 distance. It will be a liberal assumption to suppose it was ^0 = 0.1, for then 

 the initial perigee and apogee distances differed by 1,800 miles. With this 

 value and putting D = P we find as the smaller root of the first equation 

 of (98) Do = 0.205797^ = 4.939 hours. This corresponds to an initial mean 

 distance of 9,207.7 miles, not differing materially from that found when the 

 initial eccentricity was neglected. 



