AND ON THE REMOTE HISTORY OF THE EARTH. 
505 
Combining these values by the rules of the calculus of finite differences, I find 
{=13° 59'. 
This final value of £ (viz.: *5179) makes the moon’s sidereal period 12 hours, and 
the value of N (viz.: 1 *321) makes the day 5 hours 55 minutes. 
These results complete the integration of the fifth period. 
The physical meaning of the results for all five periods is given in the following 
table:— 
Sidereal day in m.s. 
hours and minutes. 
Moon’s sidereal period 
in m.s. days. 
Obliquity ef 
ecliptic. 
h. m. 
Initial 23 56 
27'32 days 
23° 28' • 
15 28 
18-62 „ 
20° 28' 
9 55 
8-17 „ 
17° 4' 
7 49 
3'59 ,, 
15° 22' * 
Final 5 55 
12 hours 
14° 0' * 
It is 'worthy of notice that at the end of the first period there were 28’9 days of 
that time in the then sidereal month ; whilst at the end of the second period there 
were only 19*7. It seems then that at the present time tidal friction has, in a sense, 
done more than half its work, and that the number of days in the month has passed 
its maximum on its way towards the state of things in which the day and month are 
of equal length—as investigated in the following section. 
In the last column of the preceding table the last two results in the column giving 
the obliquity of the ecliptic (which are marked with asterisks) cannot safely be 
accepted, because, as I have reason to believe, the simultaneous changes of inclination 
of the lunar orbit will, after the end of the second period of integration, have begun 
to influence the results perceptibly. 
For this same reason the integration, which has been carried to the critical point 
cli 
where n cos i— 2/2, and where —changes sign, will not be pursued any further. Never¬ 
theless we shall be able to trace the moon’s periodic time, and the length of day to 
their initial condition. It is obvious that as long as n is greater than SI, there will 
be tidal friction, and n will continue to approach SI, whilst both increase retrospectively 
in magnitude. 
I shall now refer to a critical phase in the relationship between n and SI, of a totally 
different character from the preceding one, and which must occur at a point a little 
more remote in time than that at which the above integration stops. 
This critical phase occurs when the free nutation of the oblate spheroid has a fre¬ 
quency equal to that of the forced fortnightly nutation. 
In the ordinary theory of the precession and nutation of a rigid oblate spheroid, the 
fortnightly nutation arises out of terms in the couples acting about a pair of axes 
fixed in the equator, which have speeds n — 2 SI and n-j-2/2. If C and A be the 
greatest and least principal moments of inertia, then on integration these terms are 
3 T 2 
