' APPLICATION OF CASE XII TO THE EARTH-MOON SYSTEM. 119 



XIII. APPLICATION OF CASE XII TO THE EARTH- 

 MOON SYSTEM. 



There is a difficulty in attempting to determine the position of the 

 invariable plane of the earth-moon system, for the plane of the moon's 

 orbit is continually changed by the perturbative action of the sun. In 

 the course of about 9^ years the line of nodes of the moon's orbit makes a 

 complete revolution, and the inclination of the plane of the earth's equator 

 varies all the way from 23.5° -f 7° to 23.5°-7°. We shall suppose here 

 that the sun is not disturbing the orbit of the moon, and that the inclina- 

 tion of its orbit to the plane of the ecliptic is zero. Let us take the plane 

 of the ecliptic as the original a:?/-plane. Then the angle between the plane 

 of the ecliptic and the invariable plane is given by 



. M Pi +^- cos (23.5°) 



COS 1 ^ — - = 



V M' + M'^ + M"^- ^p,^2jn^ ,„, (23.5°) + 5^' 



Since Qi=Q-\-7: we have 



i, = 23.5°-i (121) 



In the case of the earth-moon system we find 



i: = 4° t\ = 19.5° (122) 



Then M of (116) becomes M = 3.59654. With this value of M the smallest 

 root of (117) is P = 0.20852. This period corresponds to a distance of 

 72 = 9,364 miles. However, the initial inclinations could not have been 

 precisely zero, and consequently the initial distance must have been some- 

 what less than this amount. But the chief point of interest is that the 

 factors neglected in the discussion of section V so far all make the initial 

 distance greater than that found in that place. 



