824 
MR. G. H. DARWIN ON THE SECULAR CHANGES IN 
If the integration is to be carried still farther back, the solar action may henceforth 
be neglected, and the motion may be referred to the invariable plane of the system. 
This plane undergoes a precessional motion due to the sun, which will not interfere 
with the treatment of it as though fixed. It is inclined to the ecliptic at about 
11° 45', because, at the time when we suppose the solar action to cease, the moment 
of momentum of the earth’s rotation is larger than that of orbital motion, and therefore 
the earth’s proper plane represents the invariable plane of the system more nearly than 
does the moon’s proper plane. 
The inclination i of the equator to the invariable plane must be taken as about 3°, 
and j that of the lunar orbit as something like 5° 30'. The ratio of the two angles 
5° 30' and 3° must be equal to 1;84, which is in, the ratio of the moment of momentum 
of the earth’s rotation to that of orbital motion, at the point where the preceding 
integration ceases. 
Then in the more remote past the angle i will continue to diminish, until the point 
is reached where the moon’s period is about 12 hours and that of the earth’s rotation 
about 6 hours. The angle j will continue increasing at an accelerating rate. 
This may be shown as follows :— 
The equations of motion are now those of Part II., which may be written 
kn % = 
But since i/j=g/hi= l/lll, they become 
hl d% lo o tan 24 = 
log tan & = 
l + m 
■- 4 ’ 
m b 
(i-f m)d 
(Compare with the first of equations (255) given in Part III., when r'=0.) 
These equations are not independent, because of the relationship which must always 
subsist between i and j. 
Then substituting from (251) we have for the case of small viscosity 
hn d £ lo g tan bj- 
log tan 
O 
¥ 
l + m 
2(i-xj 
(l + m)(l—2\) 
2(1—A) ~ 
