AND ON THE REMOTE HISTORY OF THE EARTH. 
489 
it is a question of days or weeks, whilst in the variations of the amplitudes and 
frequencies of the inequalities it is a question of millions of years. Hence the above 
method is safely applicable here. 
It is worthy of remark that it has been nowhere assumed that the amplitudes of the 
periodic inequalities are small compared with the non-periodic parts of the expression. 
A precisely similar argument will be applicable to every case where occasion will 
arise to change the independent variable. The change will accordingly be carried out 
without further comment, it being always understood that both dependent and inde¬ 
pendent variable are the average values of the quantities for which their symbols 
would in general stand.* 
Then dividing (69) by (70) we have 
Now y = 
A 
. sin 2e' 
W 
sin 4e 
sin 2e 
= sir i 
, + X 
sin 4e 
approximately. 
This approximation will be suffi¬ 
ciently accurate, because the last term is small and is diminishing. For the same 
reason, only a small error will be incurred by treating it as constant, provided the 
integration be not carried over too large a field—a condition satisfied by the proposed 
“ periods of integration.” Attribute then to i, e, e' average values, and put 
y— 
, t. . 0 . sin 2e' 
j— Slll~ l ——— 
t 0 sm 4e 
and integrate. Then we have 
(72) 
ifci+Ma-A+^i-PRyU-f 7 )}.(is) 
This is the approximate form of the equation of conservation of moment of momentum, 
and it is very nearly accurate, provided f does not vary too widely. 
By putting /3= 0, y=0, we see that the equation is independent of the obliquity, if 
there be only two bodies, the earth and moon, provided we neglect the fourth power of 
the sine of the obliquity. 
The equation of reaction (68) may be written 
fit o / 2 
(W sin 4e+Xsin 2e) .(74) 
g 
* In order to feel complete confidence in my view, I placed the question before Mr. E. ,T. Routh, and 
with great kindness he sent me some remarks on the subject, in which he confirmed the correctness of 
my pi’ocedure, although he arrived at the conclusion from rather a different point of view. 
3 r 2 
