96 THE TIDAL PROBLEM. 



Since M is greater than this number it follows that (28) has two real roots. 

 It is easily found by approximation processes that the roots of (28) are ^ 



Pi =0.20535 days =4.9284 hours P2 = 47.705 days (46) 



Since the present value of P lies between these limits it must always remain 

 between them and continually approach Pj. From the formula 



_ 27ra' 



where k is the Gaussian constant, and the units are the mean solar day, 

 r', and S, we find that the distances R^ and R2 corresponding to Pj and P^ 



*'® Pi =9, 194.35 miles P2= 345,355 miles (47) 



The maximum possible number of days in a month is at once found 



from (39) to be iV = 29.559 (48) 



The corresponding length of the month expressed in terms of present 

 mean solar days is, from the first equation of (39), 



P = 20.345 (49) 



Since the month is increasing and now greater than 20.3 days, the system 

 is already beyond the condition of maximum number of days in a month. 

 Let us apply these results to Darwin's hypothesis of the separation of 

 the moon from the earth by fission, remembering, of course, that a number 

 of factors involved in the actual case have been omitted. At the time of 

 separation their periods of rotation and of revolution about their center of 

 gravity must have been equal. But the solution shows that they moved as 

 a rigid body when the surface of the moon was 9,194.4— (3,958.2 + 1,081.5) 

 = 4,154.7 miles from the surface of the earth, which contradicts the hypoth- 

 esis that they had just separated by fission. But this is neglecting the 

 earth's oblateness, which must have been great. To get an idea of the 

 possibilities let us examine a number of modifying hypotheses. First let 

 us suppose that the earth was then so oblate that its equator reached to 

 the moon, and that the law of density was such as to keep its moment of 

 momentum and volume unchanged. Then we find that 



equatorial radius = 8, 112.9 miles polar radius = 942.2 miles 



Obviously the spheroid would have broken up long before it attained this 

 degree of oblateness, and under the hypothesis that the moment of momen- 

 tum was as it would have been in a sphere the equatorial zone would have 

 been so rare that one could not account for the matter in the moon, f^} 



In order to avoid the difficulty of the rare periphery, forced by the 

 condition on the moment of momentum, we may waive this condition. 

 Assuming simply that the earth was oblate, let us find the qualitative 

 effects on the initial distance of the moon. The moment of momentum for 



» Darwin in 2, p. 508, taking the earth as a homogeneous spheroid and other^ data 

 somewhat different, foxmd F, ^^5.6 h., P2=:55.5 d. 



