APPLICATION OF SECTION VII TO THE EARTH-MOON SYSTEM. 105 



VIII. APPLICATION OF SECTION VII TO THE EARTH- 

 MOON SYSTEM. 



With the data given in (40) we find 



(-) »^2 ==/cmi =0.000914 Wi (73) 



Consequently the moment of momentum and energy of the earth-moon 

 system are respectively 



M = 3.01 187 + 0.62803 + 0.00002 = 3.63992 

 E = -0.34632 + 1.95689 + 0.00001 =1.61058 . 



(74) 



The limiting value of M for which a maximum or minimum may occur is, 

 by (65), 1.5614, Since the actual M is greater than this quantity there 

 are a minimax and a minimum. 



We shall now solve (61) and find the value of u corresponding to the 

 minimax. With the values of « and M given in (73) and (74) we find by 

 methods of approximation that the value of u satisfying (61) and corre- 

 sponding to the minimax is 



w =3.04719 ti;= 2m =6.09438 (75) 



The corresponding common period of the system and distance of the moon 

 are respectively 



Pi =0.20554 day =4.9329 hours T^i =9,200 miles (76) 



or more than 42 miles greater than when the moon's rotational momentum 

 and energy were neglected. The whole energy of the system for this value 



"'"-"'^ £ = 10.666 (77) 



At the present time in the earth-moon system we find from the data 

 of (40) and (73) and the condition D^ = P that 



w =0.6280257 v =0.0000005 w=w+v =0.6280262 (78) 



Since the present energy, given in the second equation of (74), is less than 

 that corresponding to A of fig. 13, which is given in (77), and since the 

 present value of w given in (78) is less than that corresponding to A, given 

 in (75), the energy will continually approach the value corresponding to 

 the minimum B, when the earth-moon system will move as a rigid body. 

 Although it is thus possible, under the hypotheses, to draw positive con- 

 clusions as to the conditions toward which the system is tending, it is, of 

 course, not possible to affirm, even aside from all factors neglected in this 

 section, that the system ever descended from the condition corresponding 

 to A. This discussion simply gives the numerical values belonging to the 

 condition corresponding to A, which may or may not have been verified. 

 Thus the real question of interest gets no conclusive answer by supposing 

 the moon of finite size; the introduction of this factor simply embarrasses 

 the fission theory a little more. 



